If you find this useful or would be interested in providing feedback and suggestions, please let me know at dwharder@gmail.com. See my other related pages on nice eigenvalue decompositions (eigendecompositions), integer matrices with integer eigenvalues, integer matrices with integer singular values, integer-normed vectors, integer-normed complex vectors, geometric sequences with nice norms.

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When looking for examples in linear algebra courses, it is often preferable to have examples that have nice representations on the blackboard. Consequently, if the norm contains a surd, it becomes difficult to work with, and these examples are problematic in the class room. Here are integer vectors that have integer norms. In some cases, it is possible to have orthonormal bases using ratios of simple vectors; these, too, are presented.

• Two-dimensional vectors
• Generating mutually-orthogonal two-dimensional vectors
• Normalized two-dimensional vectors with terminating decimal representation
• Other two-dimensional integer vectors with integer norms
• Three-dimensional vectors
• Generating mutually-orthogonal three-dimensional vectors
• Normalized three-dimensional vectors with terminating decimal representation
• Other three-dimensional integer vectors with integer norms
• Four-dimensional vectors
• Generating mutually-orthogonal four-dimensional vectors
• Other mutually-orthogonal four-dimensional vectors
• Normalized four-dimensional vectors with terminating decimal representation
• Other four-dimensional integer vectors with integer norms
• Five-dimensional vectors
• Normalized five-dimensional vectors with terminating decimal representation
• Other five-dimensional integer vectors with integer norms
• Six-dimensional vectors
• Normalized six-dimensional vectors with terminating decimal representation
• Other six-dimensional integer vectors with integer norms

Two-dimensional vectors

Generating mutually-orthogonal two-dimensional vectors

Given any one vector $\begin{pmatrix} \alpha \\ \beta \end{pmatrix}$ in ${\bf R}^2$, it is always possible to construct two mutually orthogonal vectors as follows:

$\begin{pmatrix} \alpha \\ \beta \end{pmatrix}$ and either $\begin{pmatrix} \beta \\ -\alpha \end{pmatrix}$ or $\begin{pmatrix} -\beta \\ \alpha \end{pmatrix}$

Therefore, if the first vector is normalized, so is the second. If normalized, an orthogonal matrix therefore, is

$\begin{pmatrix} \alpha & \beta\\ \beta & -\alpha \end{pmatrix}$ or $\begin{pmatrix} \alpha & \beta\\ -\beta & \alpha \end{pmatrix}$

Normalized two-dimensional vectors with terminating decimal representation

Vector	Normalized vector	Norm
[  3    4]	[0.6     0.8]	5
[  7   24]	[0.28    0.96]	25
[ 44  117]	[0.352   0.936]	125
[336  527]	[0.5376  0.8432]	625
[237 3116]	[0.07584 0.99712]	3125

Other two-dimensional integer vectors with integer norms

All 2-dimensional integer vectors with integer 2-norms less than $100$ are:

Vector	Norm
[ 5 12]	13
[ 8 15]	17
[20 21]	29
[12 35]	37
[ 9 40]	41
[28 45]	53
[11 60]	61
[16 63]	65
[33 56]	65
[48 55]	73
[13 84]	85
[36 77]	85
[39 80]	89
[65 72]	97

Three-dimensional vectors

Generating mutually-orthogonal three-dimensional vectors

Given any two values $\alpha$ and $\beta$ it is always possible to construct three mutually orthogonal vectors as follows:

$\begin{pmatrix} \alpha \\ \beta \\ -\frac{\alpha \beta}{\alpha + \beta} \end{pmatrix}$, $\begin{pmatrix} \beta \\ -\frac{\alpha \beta}{\alpha + \beta} \\ \alpha \end{pmatrix}$, $\begin{pmatrix} -\frac{\alpha \beta}{\alpha + \beta} \\ \alpha \\ \beta \end{pmatrix}$

Therefore, if the first vector is normalized, so are the second and third. If normalized, the following, therefore, is a symmetric orthogonal matrix:

$\begin{pmatrix} \alpha & \beta & \frac{\alpha \beta}{\alpha + \beta} \\ \beta & -\frac{\alpha \beta}{\alpha + \beta} & -\alpha \\ \frac{\alpha \beta}{\alpha + \beta} & -\alpha & \beta \end{pmatrix}$.

Swapping any two columns will produce a non-symmetric matrix (assuming these three values are different). Any row or column can be multipled by $-1$.

All such integer vectors that have a 2-norm less than $100$ are:

$\left( \alpha, \beta, \frac{\alpha \beta}{\alpha + \beta}\right)$	Norm
[ 2  2  1]	3
[ 3  6  2]	7
[ 4 12  3]	13
[10 15  6]	19
[ 5 20  4]	21
[ 6 30  5]	31
[21 28 12]	37
[14 35 10]	39
[ 7 42  6]	43
[24 40 15]	49
[ 8 56  7]	57
[36 45 20]	61
[18 63 14]	67
[ 9 72  8]	73
[30 70 21]	79
[10 90  9]	91
[55 66 30]	91
[44 77 28]	93
[33 88 24]	97

If one vector is normalized, so are all, so one such set is:

$\begin{pmatrix} \frac{2}{3} \\ \frac{2}{3} \\ \frac{1}{3} \end{pmatrix}$, $\begin{pmatrix} \frac{2}{3} \\ -\frac{1}{3} \\ -\frac{2}{3} \end{pmatrix}$, $\begin{pmatrix} \frac{1}{3} \\ -\frac{2}{3} \\ \frac{2}{3} \end{pmatrix}$

No such triplet have terminating decimal representations when normalized. The following is a corresponding orthogonal matrix:

$\frac{1}{3}\begin{pmatrix} 2 & 2 & 1 \\ 2 & -1 & -2 \\ 1 & -2 & 2 \end{pmatrix}$.

Normalized three-dimensional vectors with terminating decimal representation

Vector	Normalized vector	Norm
[ 9 12  20]	[0.36  0.48  0.8]	25
[12 15  16]	[0.48  0.6   0.64]	25
[ 3 80  96]	[0.024 0.64  0.768]	125
[12 45 116]	[0.096 0.64  0.768]	125
[12 60 109]	[0.096 0.48  0.872]	125
[19 60 108]	[0.152 0.48  0.864]	125
[21 28 120]	[0.168 0.224 0.96]	125
[21 72 100]	[0.128 0.576 0.8]	125
[24 60 107]	[0.192 0.48  0.856]	125
[24 80  93]	[0.192 0.64  0.744]	125
[28 75  96]	[0.224 0.6   0.768]	125
[35 72  96]	[0.28  0.576 0.768]	125
[44 45 108]	[0.362 0.36  0.864]	125
[53 60  96]	[0.424 0.48  0.768]	125

Other three-dimensional integer vectors with integer norms

Vector	Norm
[1  2  2]	3
[2  3  6]	7
[1  4  8]	9
[4  4  7]	9
[2  6  9]	11
[6  6  7]	11
[3  4 12]	13
[2  5 14]	15
[2 10 11]	15
[1 12 12]	17
[8  9 12]	17
[6 10 15]	19
[1 12 12]	17
[8  9 12]	17
[6 10 15]	19

Four-dimensional vectors

Generating mutually-orthogonal four-dimensional vectors

Given any one vector $\begin{pmatrix} \alpha \\ \beta \\ \gamma \\ \delta \end{pmatrix}$ in ${\bf R}^4$, it is always possible to construct four mutually orthogonal vectors as follows:

$\begin{pmatrix} \alpha \\ \beta \\ \gamma \\ \delta \end{pmatrix}$, $\begin{pmatrix} \beta \\ -\alpha \\ \delta \\ -\gamma \end{pmatrix}$, $\begin{pmatrix} \gamma \\ -\delta \\ -\alpha \\ \beta \end{pmatrix}$, $\begin{pmatrix} \delta \\ \gamma \\ -\beta \\ -\alpha \end{pmatrix}$

Therefore, if the first vector is normalized, then all four vectors will be normalize.

Other mutually-orthogonal four-dimensional vectors

While the above generates a set of orthogonal vectors, the entries are simply permutations and negations of the entries of the first vector. Here are other vectors that are not simple permutations.

[0.5  0.5  0.1  0.7 
 0.5  0.5 -0.1 -0.7 
 0.5 -0.5 -0.7  0.1 
 0.5 -0.5  0.7 -0.1]

[0.4  0.8  0.44  0.08 
 0.4  0.2 -0.8   0.4  
 0.8 -0.4  0.08 -0.44 
 0.2 -0.4  0.4   0.8 ]

[0.4   0.4   0.2   0.8  
 0.4  -0.4   0.8  -0.2  
 0.52  0.64 -0.08 -0.56 
 0.64 -0.52 -0.56  0.08]

[0.4   0.4   0.2   0.8  
 0.4   0.4  -0.8  -0.2  
 0.52  0.32  0.56 -0.56 
 0.64 -0.76 -0.08  0.08]

Normalized four-dimensional vectors with terminating decimal representation

These are all vectors in ${\bf R}^4$ that have norms of 1, 5, 10 or 25 (up to permutations and negation) where the norm is less than or equal to 25, and thus, these are the only integer vectors in ${\bf R}^4$ with such magnitudes that have normalized variations that have terminating decimal representations.

Vector	Normalized vector	Norm
[ 1  1  1  1]	[0.5  0.5  0.5  0.5]	2
[ 1  2  2  4]	[0.2  0.4  0.4  0.8]	5
[ 1  1  7  7]	[0.1  0.1  0.7  0.7]	10
[ 1  5  5  7]	[0.1  0.5  0.5  0.7]	10
[ 1  3  3  9]	[0.1  0.3  0.3  0.9]	10
[ 2  2 16 19]	[0.08 0.08 0.64 0.76]	25
[ 2  3  6 24]	[0.08 0.12 0.24 0.96]	25
[ 2  4 11 22]	[0.08 0.16 0.44 0.88]	25
[ 2  5 14 20]	[0.08 0.2  0.56 0.8]	25
[ 2  6 12 21]	[0.08 0.24 0.48 0.84]	25
[ 2  8 14 19]	[0.08 0.32 0.56 0.76]	25
[ 2 10 11 20]	[0.08 0.4  0.44 0.8]	25
[ 2 13 14 16]	[0.08 0.52 0.56 0.64]	25
[ 3  6 16 28]	[0.12 0.24 0.64 0.72]	25
[ 4  4  8 23]	[0.16 0.16 0.32 0.92]	25
[ 4  5 10 22]	[0.16 0.2  0.4  0.88]	25
[ 4 16 17 25]	[0.16 0.32 0.64 0.68]	25
[ 6 11 12 18]	[0.24 0.44 0.48 0.72]	25
[ 7  8 16 16]	[0.28 0.32 0.64 0.64]	25
[ 8 10 10 19]	[0.32 0.4  0.4  0.76]	25
[ 8 13 14 14]	[0.32 0.52 0.56 0.56]	25
[ 9 12 12 16]	[0.36 0.48 0.48 0.64]	25
[10 10 13 16]	[0.4  0.4  0.48 0.64]	25

Other four-dimensional integer vectors with integer norms

All other integer vectors with integer norms less than or equal to 20 (up to permutations and signs) are are listed here grouped by the 2-norms.

Vector	Norm
[1  1  3  5]	6
[1  4  4  4]	7
[2  2  4  5]	7
[2  2  3  8]	9
[2  4  5  6]	9
[1  2  4 10]	11
[2  2  7  8]	11
[4  4  5  8]	11
[1  2  8 10]	13
[2  4  7 10]	13
[4  4  4 11]	13
[4  5  8  8]	13
[4  6  6  9]	13
[1  1  5 13]	14
[1  5  7 11]	14
[3  3  3 13]	14
[3  5  9  9]	14
[5  5  5 11]	14
[1  4  8 12]	15
[2  3  4 14]	15
[2  4  6 13]	15
[2  6  8 11]	15
[3  4 10 10]	15
[4  4  7 12]	15
[4  8  8  9]	15
[5  6  8 10]	15
[1  4  4 16]	17
[2  2  5 16]	17
[2  4 10 13]	17
[2  5  8 14]	17
[2  8 10 11]	17
[3  6 10 12]	17
[5  8 10 10]	17
[1  3  5 17]	18
[1  7  7 15]	18
[1  9 11 11]	18
[3  5 11 13]	18
[5  5  7 15]	18
[5  7  9 13]	18
[1  2 10 16]	19
[1  8 10 14]	19
[2  2  8 17]	19
[3  8 12 12]	19
[4  5  8 16]	19
[4  7 10 14]	19
[6  6  8 15]	19
[6  9 10 12]	19
[8  8  8 13]	19

Five-dimensional vectors

Normalized five-dimensional vectors with terminating decimal representation

Each of these integer vectors, when divided by the norm, has a terminating decimal representation. The vectors are grouped by the 2-norm of the integer vector, with the integer vector first and the normalized vector second.

Vector	Normalized vector	Norm
[1  1  1  2  3]	[0.25 0.25 0.25 0.5 0.75]	4
[2  1  1  2  3]	[0.4 0.4 0.4 0.4 0.6]	5
[1  1  1  5  6]	[0.125 0.125 0.125 0.625 0.75]	8
[1  1  2  3  7]	[0.125 0.125 0.25  0.375 0.875]	8
[1  2  3  5  5]	[0.125 0.25  0.375 0.625 0.625]	8
[1  3  3  3  6]	[0.125 0.375 0.375 0.375 0.75]	8
[1  1  1  4  9]	[0.1 0.1 0.1 0.4 0.9]	10
[1  1  3  5  8]	[0.1 0.1 0.3 0.5 0.8]	10
[1  3  4  5  7]	[0.1 0.3 0.4 0.5 0.7]	10
[3  3  3  3  8]	[0.3 0.3 0.3 0.3 0.8]	10
[3  4  5  5  5]	[0.3 0.4 0.5 0.5 0.5]	10
[1  1  2  5 15]	[0.0625 0.0625 0.125  0.3125 0.9375]	16
[1  1  2  9 13]	[0.0625 0.0625 0.125  0.5625 0.8125]	16
[3  3  6  9 11]	[0.1875 0.1875 0.375  0.5625 0.6875]	16
[2  5  5  9 11]	[0.125  0.3125 0.3125 0.5625 0.6875]	16
[2  3  9  9  9]	[0.125  0.1875 0.5625 0.5625 0.5625]	16
[2  3  5  7 13]	[0.125  0.1875 0.3125 0.4375 0.8125]	16
[2  3  3  3 15]	[0.125  0.1875 0.1875 0.1875 0.9375]	16
[1  6  7  7 11]	[0.0625 0.375  0.4375 0.4375 0.6875]	16
[1  5  7  9 10]	[0.0625 0.3125 0.4375 0.5625 0.625]	16
[1  5  5  6 13]	[0.0625 0.3125 0.3125 0.375  0.8125]	16
[1  3  5 10 11]	[0.0625 0.1875 0.3125 0.625  0.6875]	16
[1  3  5  5 14]	[0.0625 0.1875 0.3125 0.3125 0.875]	16
[1  2  7  9 11]	[0.0625 0.125  0.4375 0.5625 0.6875]	16
[1  2  3 11 11]	[0.0625 0.125  0.1875 0.6875 0.6875]	16
[1  1  6  7 13]	[0.0625 0.0625 0.375  0.4375 0.8125]	16
[1  1  3  7 14]	[0.0625 0.0625 0.1875 0.4375 0.875]	16
[5  5  6  7 11]	[0.3125 0.3125 0.375  0.4375 0.6875]	16
[5  5  5  9 10]	[0.3125 0.3125 0.3125 0.5625 0.625]	16
[3  7  7  7 10]	[0.1875 0.4375 0.4375 0.4375 0.625]	16
[3  6  7  9  9]	[0.1875 0.375  0.4375 0.5625 0.5625]	16
[2  5  5 11 15]	[0.1  0.25 0.25 0.55 0.75]	20
[1  1  1  6 19]	[0.05 0.05 0.05 0.3  0.95]	20
[7  7  9 10 11]	[0.35 0.35 0.45 0.5  0.55]	20
[6  9  9  9 11]	[0.3  0.45 0.45 0.45 0.55]	20
[5  7  7  9 14]	[0.25 0.35 0.35 0.45 0.7]	20
[5  6  7 11 13]	[0.25 0.3  0.35 0.55 0.65]	20
[5  5  9 10 13]	[0.25 0.25 0.45 0.5  0.65]	20
[5  5  5  6 17]	[0.25 0.25 0.25 0.3  0.85]	20
[3  7 10 11 11]	[0.15 0.35 0.5  0.55 0.55]	20
[3  6  7  9 15]	[0.15 0.3  0.35 0.45 0.75]	20
[3  5  7 11 14]	[0.15 0.25 0.35 0.55 0.7]	20
[3  3  6 11 15]	[0.15 0.15 0.3  0.55 0.75]	20
[3  3  3  7 18]	[0.15 0.15 0.15 0.35 0.9]	20
[2  5  9 11 13]	[0.1  0.25 0.45 0.55 0.65]	20
[2  3  9  9 15]	[0.1  0.15 0.45 0.45 0.75]	20
[2  3  7  7 17]	[0.1  0.15 0.35 0.35 0.85]	20
[2  3  7 13 13]	[0.1  0.15 0.35 0.65 0.65]	20
[1  7  9 10 13]	[0.05 0.35 0.45 0.5  0.65]	20
[1  6 11 11 11]	[0.05 0.3  0.55 0.55 0.55]	20
[1  5  7 10 15]	[0.05 0.25 0.35 0.5  0.75]	20
[1  5  6  7 17]	[0.05 0.25 0.3  0.35 0.85]	20
[1  5  5  5 18]	[0.05 0.25 0.25 0.25 0.9]	20
[1  3 10 11 13]	[0.05 0.15 0.5  0.55 0.65]	20
[1  3  5 13 14]	[0.05 0.15 0.25 0.65 0.7]	20
[1  2  7 11 15]	[0.05 0.1  0.35 0.55 0.75]	20
[1  2  5  9 17]	[0.05 0.1  0.25 0.45 0.85]	20
[1  2  3  5 19]	[0.05 0.1  0.15 0.25 0.95]	20
[1  1  9 11 14]	[0.05 0.05 0.45 0.55 0.7]	20
[1  1  5  7 18]	[0.05 0.05 0.25 0.35 0.9]	20
[1  1  3 10 17]	[0.05 0.05 0.15 0.5  0.85]	20
[1  1  2 13 15]	[0.05 0.05 0.1  0.65 0.75]	20
[1  5  6 13 13]	[0.05 0.25 0.3  0.65 0.65]	20
[2  6  6 15 18]	[0.08 0.24 0.24 0.6  0.72]	25
[2  5 12 14 16]	[0.08 0.2  0.48 0.56 0.64]	25
[2  4 10 12 19]	[0.08 0.16 0.4  0.48 0.76]	25
[2  4  6 13 20]	[0.08 0.16 0.24 0.52 0.8]	25
[2  4  5 16 18]	[0.08 0.16 0.2  0.64 0.72]	25
[2  3 12 12 18]	[0.08 0.12 0.48 0.48 0.72]	25
[2  3 10 16 16]	[0.08 0.12 0.4  0.64 0.64]	25
[2  3  4 14 20]	[0.08 0.12 0.16 0.56 0.8]	25
[2  2 14 14 15]	[0.08 0.08 0.56 0.56 0.6]	25
[1 10 10 10 18]	[0.04 0.4  0.4  0.4  0.72]	25
[1  6 14 14 14]	[0.04 0.24 0.56 0.56 0.56]	25
[1  4  8 12 20]	[0.04 0.16 0.32 0.48 0.8]	25
[1  2 10 14 18]	[0.04 0.08 0.4  0.56 0.72]	25
[7 11 13 18 19]	[0.21875 0.34375 0.40625 0.5625  0.59375]	32
[7 10 15 17 19]	[0.21875 0.3125  0.46875 0.53125 0.59375]	32
[6 11 17 17 17]	[0.1875  0.34375 0.53125 0.53125 0.53125]	32
[6  7 17 17 19]	[0.1875  0.21875 0.53125 0.53125 0.59375]	32
[5 15 15 15 18]	[0.15625 0.46875 0.46875 0.46875 0.5625]	32
[5 14 15 17 17]	[0.15625 0.4375  0.46875 0.53125 0.53125]	32
[5  9 14 19 19]	[0.15625 0.28125 0.4375  0.59375 0.59375]	32
[5  5 17 18 19]	[0.15625 0.15625 0.53125 0.5625  0.59375]	32
[3 13 14 17 19]	[0.09375 0.40625 0.4375  0.53125 0.59375]	32
[2  9 17 17 19]	[0.0625  0.28125 0.53125 0.53125 0.59375]	32
[2  3 17 19 19]	[0.0625  0.09375 0.53125 0.59375 0.59375]	32
[1 13 13 18 19]	[0.03125 0.40625 0.40625 0.5625  0.59375]	32
[1 11 17 17 18]	[0.03125 0.34375 0.53125 0.53125 0.5625]	32
[1  7 17 18 19]	[0.03125 0.21875 0.53125 0.5625  0.59375]	32
[6 13 13 17 19]	[0.1875  0.40625 0.40625 0.53125 0.59375]	32

Other five-dimensional integer vectors with integer norms

These are all such integer vectors that have norm less than or equal to 19. These integer vectors are grouped by the 2-norms.

Vector	Norm
[1  1  3  3  4]	6
[1  2  2  2  6]	7
[2  2  3  4  4]	7
[2  3  4  4  6]	9
[1  2  2  6  6]	9
[1  2  4  6  8]	11
[2  2  2  3 10]	11
[2  2  4  4  9]	11
[2  3  6  6  6]	11
[2  4  4  6  7]	11
[3  4  4  4  8]	11
[1  1  5  6  9]	12
[1  2  3  3 11]	12
[1  2  3  7  9]	12
[1  3  3  5 10]	12
[1  3  6  7  7]	12
[2  3  5  5  9]	12
[3  5  5  6  7]	12
[1  2  2  4 12]	13
[1  2  6  8  8]	13
[1  4  4  6 10]	13
[2  2  2  6 11]	13
[2  2  4  8  9]	13
[2  2  5  6 10]	13
[2  4  6  7  8]	13
[3  4  4  8  8]	13
[5  6  6  6  6]	13
[1  1  1  7 12]	14
[1  1  3  4 13]	14
[1  1  3  8 11]	14
[1  1  5  5 12]	14
[1  1  7  8  9]	14
[1  3  4  7 11]	14
[1  4  7  7  9]	14
[1  5  5  8  9]	14
[3  3  3  5 12]	14
[3  3  4  9  9]	14
[3  4  5  5 11]	14
[3  5  7  7  8]	14
[4  5  5  7  9]	14
[2  2  3  8 12]	15
[2  2  6  9 10]	15
[2  4  5  6 12]	15
[2  6  6  7 10]	15
[3  4  6  8 10]	15
[4  4  6  6 11]	15
[5  6  6  8  8]	15
[1  4  8  8 12]	17
[2  2  2  9 14]	17
[2  2  3  4 16]	17
[2  2  4 11 12]	17
[2  2  6  7 14]	17
[2  2  9 10 10]	17
[2  3  4  8 14]	17
[2  4  5 10 12]	17
[2  4  6  8 13]	17
[2  6  7 10 10]	17
[2  6  8  8 11]	17
[3  4  8 10 10]	17
[3  6  6  8 12]	17
[4  4  4  4 15]	17
[4  4  5  6 14]	17
[4  4  6 10 11]	17
[4  4  7  8 12]	17
[4  8  8  8  9]	17
[5  6  8  8 10]	17
[6  6  6  9 10]	17
[1  1  3 12 13]	18
[1  1  4  9 15]	18
[1  3  3  4 17]	18
[1  3  3  7 16]	18
[1  3  5  8 15]	18
[1  3  7 11 12]	18
[1  3  8  9 13]	18
[1  7  7  9 12]	18
[3  3  4 11 13]	18
[3  3  5  5 16]	18
[3  3  8 11 11]	18
[3  4  5  7 15]	18
[3  4  7  9 13]	18
[3  5  5 11 12]	18
[3  7  8  9 11]	18
[4  5  9  9 11]	18
[5  5  7  9 12]	18
[7  7  8  9  9]	18
[1  2  4  4 18]	19
[1  2  4 12 14]	19
[1  2  6  8 16]	19
[1  4 10 10 12]	19
[1  6  6 12 12]	19
[1  6  8  8 14]	19
[2  2  2  5 18]	19
[2  2  4  9 16]	19
[2  2  6 11 14]	19
[2  2  8  8 15]	19
[2  4  4  6 17]	19
[2  4  4 10 15]	19
[2  4  6  7 16]	19
[2  4  8  9 14]	19
[2  5  6 10 14]	19
[2  6 10 10 11]	19
[2  7  8 10 12]	19
[3  4  4  8 16]	19
[4  4  4 12 13]	19
[4  4  8 11 12]	19
[4  6  7  8 14]	19
[4  8  9 10 10]	19
[5  6 10 10 10]	19
[5  8  8  8 12]	19
[6  6  8  9 12]	19

Six-dimensional vectors

Normalized six-dimensional vectors with terminating decimal representation

Each of these integer vectors, when divided by the norm, has a terminating decimal representation. The vectors are grouped by the 2-norm of the integer vector, with the integer vector first and the normalized vector second.

      [ 1  1  1  2  3  3] [0.2 0.2 0.2 0.4 0.6 0.6]                        5
      [ 1  1  1  3  4  6] [0.125 0.125 0.125 0.375 0.5 0.75]               8
      [ 1  2  3  3  4  5] [0.125 0.25 0.375 0.375 0.5 0.625]               8
      [ 1  1  1  5  6  6] [0.1 0.1 0.1 0.5 0.6 0.6]                       10
      [ 1  1  2  2  3  9] [0.1 0.1 0.2 0.2 0.3 0.9]                       10
      [ 1  1  2  3  6  7] [0.1 0.1 0.2 0.3 0.6 0.7]                       10
      [ 1  1  3  3  4  8] [0.1 0.1 0.3 0.3 0.4 0.8]                       10
      [ 1  2  3  5  5  6] [0.1 0.2 0.3 0.5 0.5 0.6]                       10
      [ 1  3  3  3  6  6] [0.1 0.3 0.3 0.3 0.6 0.6]                       10
      [ 1  3  3  4  4  7] [0.1 0.3 0.3 0.4 0.4 0.7]                       10
      [ 2  2  2  4  6  6] [0.2 0.2 0.2 0.4 0.6 0.6]                       10
      [ 2  2  3  3  5  7] [0.2 0.2 0.3 0.3 0.5 0.7]                       10
      [ 3  3  4  4  5  5] [0.3 0.3 0.4 0.4 0.5 0.5]                       10
      [ 1  1  1  3 10 12] [0.0625 0.0625 0.0625 0.1875 0.625 0.75]        16
      [ 1  1  2  3  4 15] [0.0625 0.0625 0.125 0.1875 0.25 0.9375]        16
      [ 1  1  2  5  9 12] [0.0625 0.0625 0.125 0.3125 0.5625 0.75]        16
      [ 1  1  3  8  9 10] [0.0625 0.0625 0.1875 0.5 0.5625 0.625]         16
      [ 1  1  4  6  9 11] [0.0625 0.0625 0.25 0.375 0.5625 0.6875]        16
      [ 1  1  5  6  7 12] [0.0625 0.0625 0.3125 0.375 0.4375 0.75]        16
      [ 1  2  3  3  8 13] [0.0625 0.125 0.1875 0.1875 0.5 0.8125]         16
      [ 1  2  3  7  7 12] [0.0625 0.125 0.1875 0.4375 0.4375 0.75]        16
      [ 1  2  5  8  9  9] [0.0625 0.125 0.3125 0.5 0.5625 0.5625]         16
      [ 1  3  3  4  5 14] [0.0625 0.1875 0.1875 0.25 0.3125 0.875]        16
      [ 1  3  3  4 10 11] [0.0625 0.1875 0.1875 0.25 0.625 0.6875]        16
      [ 1  3  4  5  6 13] [0.0625 0.1875 0.25 0.3125 0.375 0.8125]        16
      [ 1  3  4  7  9 10] [0.0625 0.1875 0.25 0.4375 0.5625 0.625]        16
      [ 1  3  5  6  8 11] [0.0625 0.1875 0.3125 0.375 0.5 0.6875]         16
      [ 1  5  5  5  6 12] [0.0625 0.3125 0.3125 0.3125 0.375 0.75]        16
      [ 1  5  6  7  8  9] [0.0625 0.3125 0.375 0.4375 0.5 0.5625]         16
      [ 2  2  2  6  8 12] [0.125 0.125 0.125 0.375 0.5 0.75]              16
      [ 2  3  3  3  9 12] [0.125 0.1875 0.1875 0.1875 0.5625 0.75]        16
      [ 2  3  3  4  7 13] [0.125 0.1875 0.1875 0.25 0.4375 0.8125]        16
      [ 2  3  3  7  8 11] [0.125 0.1875 0.1875 0.4375 0.5 0.6875]         16
      [ 2  3  4  5  9 11] [0.125 0.1875 0.25 0.3125 0.5625 0.6875]        16
      [ 2  3  5  5  7 12] [0.125 0.1875 0.3125 0.3125 0.4375 0.75]        16
      [ 2  3  7  7  8  9] [0.125 0.1875 0.4375 0.4375 0.5 0.5625]         16
      [ 2  4  5  7  9  9] [0.125 0.25 0.3125 0.4375 0.5625 0.5625]        16
      [ 2  4  6  6  8 10] [0.125 0.25 0.375 0.375 0.5 0.625]              16
      [ 3  3  3  6  7 12] [0.1875 0.1875 0.1875 0.375 0.4375 0.75]        16
      [ 3  3  5  7  8 10] [0.1875 0.1875 0.3125 0.4375 0.5 0.625]         16
      [ 3  4  5  5  9 10] [0.1875 0.25 0.3125 0.3125 0.5625 0.625]        16
      [ 3  4  5  6  7 11] [0.1875 0.25 0.3125 0.375 0.4375 0.6875]        16
      [ 3  6  7  7  7  8] [0.1875 0.375 0.4375 0.4375 0.4375 0.5]         16
      [ 4  5  6  7  7  9] [0.25 0.3125 0.375 0.4375 0.4375 0.5625]        16
      [ 5  5  5  6  8  9] [0.3125 0.3125 0.3125 0.375 0.5 0.5625]         16
      [ 1  1  1  3  8 18] [0.05 0.05 0.05 0.15 0.4 0.9]                   20
      [ 1  1  2  5 12 15] [0.05 0.05 0.1 0.25 0.6 0.75]                   20
      [ 1  1  2  9 12 13] [0.05 0.05 0.1 0.45 0.6 0.65]                   20
      [ 1  1  3  4  7 18] [0.05 0.05 0.15 0.2 0.35 0.9]                   20
      [ 1  1  3  6  8 17] [0.05 0.05 0.15 0.3 0.4 0.85]                   20
      [ 1  1  3  7 12 14] [0.05 0.05 0.15 0.35 0.6 0.7]                   20
      [ 1  1  3  8 10 15] [0.05 0.05 0.15 0.4 0.5 0.75]                   20
      [ 1  1  4  6 11 15] [0.05 0.05 0.2 0.3 0.55 0.75]                   20
      [ 1  1  5  6  9 16] [0.05 0.05 0.25 0.3 0.45 0.8]                   20
      [ 1  1  6  7 12 13] [0.05 0.05 0.3 0.35 0.6 0.65]                   20
      [ 1  2  3  3  4 19] [0.05 0.1 0.15 0.15 0.2 0.95]                   20
      [ 1  2  3  3 11 16] [0.05 0.1 0.15 0.15 0.55 0.8]                   20
      [ 1  2  3  4  9 17] [0.05 0.1 0.15 0.2 0.45 0.85]                   20
      [ 1  2  3  7  9 16] [0.05 0.1 0.15 0.35 0.45 0.8]                   20
      [ 1  2  3 11 11 12] [0.05 0.1 0.15 0.55 0.55 0.6]                   20
      [ 1  2  5  8  9 15] [0.05 0.1 0.25 0.4 0.45 0.75]                   20
      [ 1  2  7  9 11 12] [0.05 0.1 0.35 0.45 0.55 0.6]                   20
      [ 1  2  8  9  9 13] [0.05 0.1 0.4 0.45 0.45 0.65]                   20
      [ 1  3  3  4 13 14] [0.05 0.15 0.15 0.2 0.65 0.7]                   20
      [ 1  3  3  5 10 16] [0.05 0.15 0.15 0.25 0.5 0.8]                   20
      [ 1  3  3  8 11 14] [0.05 0.15 0.15 0.4 0.55 0.7]                   20
      [ 1  3  4  5  5 18] [0.05 0.15 0.2 0.25 0.25 0.9]                   20
      [ 1  3  4  6  7 17] [0.05 0.15 0.2 0.3 0.35 0.85]                   20
      [ 1  3  4  6 13 13] [0.05 0.15 0.2 0.3 0.65 0.65]                   20
      [ 1  3  4  7 10 15] [0.05 0.15 0.2 0.35 0.5 0.75]                   20
      [ 1  3  5  5 12 14] [0.05 0.15 0.25 0.25 0.6 0.7]                   20
      [ 1  3  5 10 11 12] [0.05 0.15 0.25 0.5 0.55 0.6]                   20
      [ 1  3  6  7  7 16] [0.05 0.15 0.3 0.35 0.35 0.8]                   20
      [ 1  3  6  8 11 13] [0.05 0.15 0.3 0.4 0.55 0.65]                   20
      [ 1  3  7  8  9 14] [0.05 0.15 0.35 0.4 0.45 0.7]                   20
      [ 1  4  5  9  9 14] [0.05 0.2 0.25 0.45 0.45 0.7]                   20
      [ 1  4  9  9 10 11] [0.05 0.2 0.45 0.45 0.5 0.55]                   20
      [ 1  5  5  6 12 13] [0.05 0.25 0.25 0.3 0.6 0.65]                   20
      [ 1  5  6  7  8 15] [0.05 0.25 0.3 0.35 0.4 0.75]                   20
      [ 1  5  7  9 10 12] [0.05 0.25 0.35 0.45 0.5 0.6]                   20
      [ 1  6  7  7 11 12] [0.05 0.3 0.35 0.35 0.55 0.6]                   20
      [ 1  6  7  8  9 13] [0.05 0.3 0.35 0.4 0.45 0.65]                   20
      [ 2  2  2 10 12 12] [0.1 0.1 0.1 0.5 0.6 0.6]                       20
      [ 2  2  4  4  6 18] [0.1 0.1 0.2 0.2 0.3 0.9]                       20
      [ 2  2  4  6 12 14] [0.1 0.1 0.2 0.3 0.6 0.7]                       20
      [ 2  2  6  6  8 16] [0.1 0.1 0.3 0.3 0.4 0.8]                       20
      [ 2  3  3  3 12 15] [0.1 0.15 0.15 0.15 0.6 0.75]                   20
      [ 2  3  3  5  8 17] [0.1 0.15 0.15 0.25 0.4 0.85]                   20
      [ 2  3  4  5 11 15] [0.1 0.15 0.2 0.25 0.55 0.75]                   20
      [ 2  3  4  9 11 13] [0.1 0.15 0.2 0.45 0.55 0.65]                   20
      [ 2  3  5  5  9 16] [0.1 0.15 0.25 0.25 0.45 0.8]                   20
      [ 2  3  5  7 12 13] [0.1 0.15 0.25 0.35 0.6 0.65]                   20
      [ 2  3  7  7  8 15] [0.1 0.15 0.35 0.35 0.4 0.75]                   20
      [ 2  3  8  9 11 11] [0.1 0.15 0.4 0.45 0.55 0.55]                   20
      [ 2  3  9  9  9 12] [0.1 0.15 0.45 0.45 0.45 0.6]                   20
      [ 2  4  5  7  9 15] [0.1 0.2 0.25 0.35 0.45 0.75]                   20
      [ 2  4  6 10 10 12] [0.1 0.2 0.3 0.5 0.5 0.6]                       20
      [ 2  4  7  9  9 13] [0.1 0.2 0.35 0.45 0.45 0.65]                   20
      [ 2  5  5  9 11 12] [0.1 0.25 0.25 0.45 0.55 0.6]                   20
      [ 2  6  6  6 12 12] [0.1 0.3 0.3 0.3 0.6 0.6]                       20
      [ 2  6  6  8  8 14] [0.1 0.3 0.3 0.4 0.4 0.7]                       20
      [ 2  7  8  9  9 11] [0.1 0.35 0.4 0.45 0.45 0.55]                   20
      [ 3  3  3  6  9 16] [0.15 0.15 0.15 0.3 0.45 0.8]                   20
      [ 3  3  4  7 11 14] [0.15 0.15 0.2 0.35 0.55 0.7]                   20
      [ 3  3  6  9 11 12] [0.15 0.15 0.3 0.45 0.55 0.6]                   20
      [ 3  3  7  8 10 13] [0.15 0.15 0.35 0.4 0.5 0.65]                   20
      [ 3  4  5  5  6 17] [0.15 0.2 0.25 0.25 0.3 0.85]                   20
      [ 3  4  5  5 10 15] [0.15 0.2 0.25 0.25 0.5 0.75]                   20
      [ 3  4  5  9 10 13] [0.15 0.2 0.25 0.45 0.5 0.65]                   20
      [ 3  4  6  7 11 13] [0.15 0.2 0.3 0.35 0.55 0.65]                   20
      [ 3  4  7  7  9 14] [0.15 0.2 0.35 0.35 0.45 0.7]                   20
      [ 3  5  5  6  7 16] [0.15 0.25 0.25 0.3 0.35 0.8]                   20
      [ 3  5  5  8  9 14] [0.15 0.25 0.25 0.4 0.45 0.7]                   20
      [ 3  5  8  9 10 11] [0.15 0.25 0.4 0.45 0.5 0.55]                   20
      [ 3  6  7  8 11 11] [0.15 0.3 0.35 0.4 0.55 0.55]                   20
      [ 3  6  7  9  9 12] [0.15 0.3 0.35 0.45 0.45 0.6]                   20
      [ 3  7  7  7 10 12] [0.15 0.35 0.35 0.35 0.5 0.6]                   20
      [ 4  4  4  8 12 12] [0.2 0.2 0.2 0.4 0.6 0.6]                       20
      [ 4  4  6  6 10 14] [0.2 0.2 0.3 0.3 0.5 0.7]                       20
      [ 4  5  6  7  7 15] [0.2 0.25 0.3 0.35 0.35 0.75]                   20
      [ 4  5  6  9 11 11] [0.2 0.25 0.3 0.45 0.55 0.55]                   20
      [ 4  6  7  7  9 13] [0.2 0.3 0.35 0.35 0.45 0.65]                   20
      [ 5  5  5  6  8 15] [0.25 0.25 0.25 0.3 0.4 0.75]                   20
      [ 5  5  5  9 10 12] [0.25 0.25 0.25 0.45 0.5 0.6]                   20
      [ 5  5  6  7 11 12] [0.25 0.25 0.3 0.35 0.55 0.6]                   20
      [ 5  5  6  8  9 13] [0.25 0.25 0.3 0.4 0.45 0.65]                   20
      [ 5  7  8  9  9 10] [0.25 0.35 0.4 0.45 0.45 0.5]                   20
      [ 6  6  8  8 10 10] [0.3 0.3 0.4 0.4 0.5 0.5]                       20
      [ 6  7  7  8  9 11] [0.3 0.35 0.35 0.4 0.45 0.55]                   20
      [ 1  1  1  3 17 18] [0.04 0.04 0.04 0.12 0.68 0.72]                 25
      [ 1  1  1  6 15 19] [0.04 0.04 0.04 0.24 0.6 0.76]                  25
      [ 1  1  2 13 15 15] [0.04 0.04 0.08 0.52 0.6 0.6]                   25
      [ 1  1  3  6 17 17] [0.04 0.04 0.12 0.24 0.68 0.68]                 25
      [ 1  1  3 10 15 17] [0.04 0.04 0.12 0.4 0.6 0.68]                   25
      [ 1  1  3 11 13 18] [0.04 0.04 0.12 0.44 0.52 0.72]                 25
      [ 1  1  5  7 15 18] [0.04 0.04 0.2 0.28 0.6 0.72]                   25
      [ 1  1  7  9 13 18] [0.04 0.04 0.28 0.36 0.52 0.72]                 25
      [ 1  1  9  9 10 19] [0.04 0.04 0.36 0.36 0.4 0.76]                  25
      [ 1  1  9 11 14 15] [0.04 0.04 0.36 0.44 0.561 0.6]                 25
      [ 1  2  2  6 16 18] [0.04 0.08 0.08 0.24 0.64 0.72]                 25
      [ 1  2  3  5 15 19] [0.04 0.08 0.12 0.2 0.6 0.76]                   25
      [ 1  2  3  9 13 19] [0.04 0.08 0.12 0.36 0.52 0.76]                 25
      [ 1  2  5  9 15 17] [0.04 0.08 0.2 0.36 0.6 0.68]                   25
      [ 1  2  6  8 14 18] [0.04 0.08 0.24 0.32 0.561 0.72]                25
      [ 1  2  7 11 15 15] [0.04 0.08 0.28 0.44 0.6 0.6]                   25
      [ 1  2  9  9 13 17] [0.04 0.08 0.36 0.36 0.52 0.68]                 25
      [ 1  3  3  7 14 19] [0.04 0.12 0.12 0.28 0.561 0.76]                25
      [ 1  3  3 11 14 17] [0.04 0.12 0.12 0.44 0.561 0.68]                25
      [ 1  3  5 13 14 15] [0.04 0.12 0.2 0.52 0.561 0.6]                  25
      [ 1  3  6  7 13 19] [0.04 0.12 0.24 0.28 0.52 0.76]                 25
      [ 1  3  6 11 13 17] [0.04 0.12 0.24 0.44 0.52 0.68]                 25
      [ 1  3  7  9 14 17] [0.04 0.12 0.28 0.36 0.561 0.68]                25
      [ 1  3  7 11 11 18] [0.04 0.12 0.28 0.44 0.44 0.72]                 25
      [ 1  3  9 13 13 14] [0.04 0.12 0.36 0.52 0.52 0.561]                25
      [ 1  3 10 11 13 15] [0.04 0.12 0.4 0.44 0.52 0.6]                   25
      [ 1  4  8 12 12 16] [0.04 0.16 0.32 0.48 0.48 0.64]                 25
      [ 1  5  5  5 15 18] [0.04 0.2 0.2 0.2 0.6 0.72]                     25
      [ 1  5  5  9 13 18] [0.04 0.2 0.2 0.36 0.52 0.72]                   25
      [ 1  5  6  7 15 17] [0.04 0.2 0.24 0.28 0.6 0.68]                   25
      [ 1  5  6  9 11 19] [0.04 0.2 0.24 0.36 0.44 0.76]                  25
      [ 1  5  6 13 13 15] [0.04 0.2 0.24 0.52 0.52 0.6]                   25
      [ 1  5  7 10 15 15] [0.04 0.2 0.28 0.4 0.6 0.6]                     25
      [ 1  6  6 10 14 16] [0.04 0.24 0.24 0.4 0.561 0.64]                 25
      [ 1  6  7  9 13 17] [0.04 0.24 0.28 0.36 0.52 0.68]                 25
      [ 1  6  8 10 10 18] [0.04 0.24 0.32 0.4 0.4 0.72]                   25
      [ 1  6  9 13 13 13] [0.04 0.24 0.36 0.52 0.52 0.52]                 25
      [ 1  6 11 11 11 15] [0.04 0.24 0.44 0.44 0.44 0.6]                  25
      [ 1  7  7  9 11 18] [0.04 0.28 0.28 0.36 0.44 0.72]                 25
      [ 1  7  9 10 13 15] [0.04 0.28 0.36 0.4 0.52 0.6]                   25
      [ 2  2  2  8 15 18] [0.08 0.08 0.08 0.32 0.6 0.72]                  25
      [ 2  2  3  8 12 20] [0.08 0.08 0.12 0.32 0.48 0.8]                  25
      [ 2  2  4  9 14 18] [0.08 0.08 0.16 0.36 0.561 0.72]                25
      [ 2  2  6  6 16 17] [0.08 0.08 0.24 0.24 0.64 0.68]                 25
      [ 2  2  6  9 10 20] [0.08 0.08 0.24 0.36 0.4 0.8]                   25
      [ 2  2  6 10 15 16] [0.08 0.08 0.24 0.4 0.6 0.64]                   25
      [ 2  2  7 10 12 18] [0.08 0.08 0.28 0.4 0.48 0.72]                  25
      [ 2  2  9 12 14 14] [0.08 0.08 0.36 0.48 0.561 0.561]               25
      [ 2  3  3  5 17 17] [0.08 0.12 0.12 0.2 0.68 0.68]                  25
      [ 2  3  3 11 11 19] [0.08 0.12 0.12 0.44 0.44 0.76]                 25
      [ 2  3  4  4 16 18] [0.08 0.12 0.16 0.16 0.64 0.72]                 25
      [ 2  3  4 12 14 16] [0.08 0.12 0.16 0.48 0.561 0.64]                25
      [ 2  3  6  8 16 16] [0.08 0.12 0.24 0.32 0.64 0.64]                 25
      [ 2  3  7  7 15 17] [0.08 0.12 0.28 0.28 0.6 0.68]                  25
      [ 2  3  7  9 11 19] [0.08 0.12 0.28 0.36 0.44 0.76]                 25
      [ 2  3  7 13 13 15] [0.08 0.12 0.28 0.52 0.52 0.6]                  25
      [ 2  3  9  9 15 15] [0.08 0.12 0.36 0.36 0.6 0.6]                   25
      [ 2  3  9 11 11 17] [0.08 0.12 0.36 0.44 0.44 0.68]                 25
      [ 2  4  4 11 12 18] [0.08 0.16 0.16 0.44 0.48 0.72]                 25
      [ 2  4  5  6 12 20] [0.08 0.16 0.2 0.24 0.48 0.8]                   25
      [ 2  4  6  7 14 18] [0.08 0.16 0.24 0.28 0.561 0.72]                25
      [ 2  4  6  8 12 19] [0.08 0.16 0.24 0.32 0.48 0.76]                 25
      [ 2  4  6 12 13 16] [0.08 0.16 0.24 0.48 0.52 0.64]                 25
      [ 2  4  9 10 10 18] [0.08 0.16 0.36 0.4 0.4 0.72]                   25
      [ 2  4 11 12 12 14] [0.08 0.16 0.44 0.48 0.48 0.561]                25
      [ 2  5  5 11 15 15] [0.08 0.2 0.2 0.44 0.6 0.6]                     25
      [ 2  5  8  8 12 18] [0.08 0.2 0.32 0.32 0.48 0.72]                  25
      [ 2  5  9 11 13 15] [0.08 0.2 0.36 0.44 0.52 0.6]                   25
      [ 2  6  6  7 10 20] [0.08 0.24 0.24 0.28 0.4 0.8]                   25
      [ 2  6  6  8 14 17] [0.08 0.24 0.24 0.32 0.561 0.68]                25
      [ 2  6  6  9 12 18] [0.08 0.24 0.24 0.36 0.48 0.72]                 25
      [ 2  6  7 12 14 14] [0.08 0.24 0.28 0.48 0.561 0.561]               25
      [ 2  6  8 10 14 15] [0.08 0.24 0.32 0.4 0.561 0.6]                  25
      [ 2  6  8 11 12 16] [0.08 0.24 0.32 0.44 0.48 0.64]                 25
      [ 2  7  7  9  9 19] [0.08 0.28 0.28 0.36 0.36 0.76]                 25
      [ 2  7  9  9 11 17] [0.08 0.28 0.36 0.36 0.44 0.68]                 25
      [ 2  8 10 12 12 13] [0.08 0.32 0.4 0.48 0.48 0.52]                  25
      [ 2  9  9 11 13 13] [0.08 0.36 0.36 0.44 0.52 0.52]                 25
      [ 2  9 10 10 12 14] [0.08 0.36 0.4 0.4 0.48 0.561]                  25
      [ 3  3  3  7 15 18] [0.12 0.12 0.12 0.28 0.6 0.72]                  25
      [ 3  3  5  5 14 19] [0.12 0.12 0.2 0.2 0.56 0.76]                   25
      [ 3  3  5 10 11 19] [0.12 0.12 0.2 0.4 0.44 0.76]                   25
      [ 3  3  6 11 15 15] [0.12 0.12 0.24 0.44 0.6 0.6]                   25
      [ 3  3  7 10 13 17] [0.12 0.12 0.28 0.4 0.52 0.68]                  25
      [ 3  3  9  9 11 18] [0.12 0.12 0.36 0.36 0.44 0.72]                 25
      [ 3  3 10 13 13 13] [0.12 0.12 0.4 0.52 0.52 0.52]                  25
      [ 3  3 11 11 13 14] [0.12 0.12 0.44 0.44 0.52 0.56]                 25
      [ 3  4  4  8 14 18] [0.12 0.16 0.16 0.32 0.56 0.72]                 25
      [ 3  4  6  8 10 20] [0.12 0.16 0.24 0.32 0.4 0.8]                   25
      [ 3  4  8 12 14 14] [0.12 0.16 0.32 0.48 0.56 0.56]                 25
      [ 3  4 10 10 12 16] [0.12 0.16 0.4 0.4 0.48 0.64]                   25
      [ 3  5  5  6 13 19] [0.12 0.2 0.2 0.24 0.52 0.76]                   25
      [ 3  5  5  9 14 17] [0.12 0.2 0.2 0.36 0.56 0.68]                   25
      [ 3  5  5 11 11 18] [0.12 0.2 0.2 0.44 0.44 0.72]                   25
      [ 3  5  7  7 13 18] [0.12 0.2 0.28 0.28 0.52 0.72]                  25
      [ 3  5  7  9 10 19] [0.12 0.2 0.28 0.36 0.4 0.76]                   25
      [ 3  5  7 11 14 15] [0.12 0.2 0.28 0.44 0.56 0.6]                   25
      [ 3  5  9 10 11 17] [0.12 0.2 0.36 0.4 0.44 0.68]                   25
      [ 3  6  6 12 12 16] [0.12 0.24 0.24 0.48 0.48 0.64]                 25
      [ 3  6  7  7 11 19] [0.12 0.24 0.28 0.28 0.44 0.76]                 25
      [ 3  6  7  9 15 15] [0.12 0.24 0.28 0.36 0.6 0.6]                   25
      [ 3  6  7 11 11 17] [0.12 0.24 0.28 0.44 0.44 0.68]                 25
      [ 3  6  8  8 14 16] [0.12 0.24 0.32 0.32 0.56 0.64]                 25
      [ 3  6 11 11 13 13] [0.12 0.24 0.44 0.44 0.52 0.52]                 25
      [ 3  7  9  9  9 18] [0.12 0.28 0.36 0.36 0.36 0.72]                 25
      [ 3  7  9 11 13 14] [0.12 0.28 0.36 0.44 0.52 0.56]                 25
      [ 3  7 10 11 11 15] [0.12 0.28 0.4 0.44 0.44 0.6]                   25
      [ 3  8  8  8 10 18] [0.12 0.32 0.32 0.32 0.4 0.72]                  25
      [ 4  4  4 12 12 17] [0.16 0.16 0.16 0.48 0.48 0.68]                 25
      [ 4  4  5 10 12 18] [0.16 0.16 0.2 0.4 0.48 0.72]                   25
      [ 4  4  6  6 11 20] [0.16 0.16 0.24 0.24 0.44 0.8]                  25
      [ 4  4  6  8 13 18] [0.16 0.16 0.24 0.32 0.52 0.72]                 25
      [ 4  4  7 12 12 16] [0.16 0.16 0.28 0.48 0.48 0.64]                 25
      [ 4  5  6  6 16 16] [0.16 0.2 0.24 0.24 0.64 0.64]                  25
      [ 4  5 10 12 12 14] [0.16 0.2 0.4 0.48 0.48 0.56]                   25
      [ 4  6  7 10 10 18] [0.16 0.24 0.28 0.4 0.4 0.72]                   25
      [ 4  6  8  8 11 18] [0.16 0.24 0.32 0.32 0.44 0.72]                 25
      [ 4  6  8 12 13 14] [0.16 0.24 0.32 0.48 0.52 0.56]                 25
      [ 4  6  9 10 14 14] [0.16 0.24 0.36 0.4 0.56 0.56]                  25
      [ 4  8  8  9 12 16] [0.16 0.32 0.32 0.36 0.48 0.64]                 25
      [ 4 10 10 11 12 12] [0.16 0.4 0.4 0.44 0.48 0.48]                   25
      [ 5  5  5  6 15 17] [0.2 0.2 0.2 0.24 0.6 0.68]                     25
      [ 5  5  5 10 15 15] [0.2 0.2 0.2 0.4 0.6 0.6]                       25
      [ 5  5  6  9 13 17] [0.2 0.2 0.24 0.36 0.52 0.68]                   25
      [ 5  5  7  9 11 18] [0.2 0.2 0.28 0.36 0.44 0.72]                   25
      [ 5  5  9 10 13 15] [0.2 0.2 0.36 0.4 0.52 0.6]                     25
      [ 5  6  6  8  8 20] [0.2 0.24 0.24 0.32 0.32 0.8]                   25
      [ 5  6  7 11 13 15] [0.2 0.24 0.28 0.44 0.52 0.6]                   25
      [ 5  6  8 10 12 16] [0.2 0.24 0.32 0.4 0.48 0.64]                   25
      [ 5  7  7  9 14 15] [0.2 0.28 0.28 0.36 0.56 0.6]                   25
      [ 5  7  9  9 10 17] [0.2 0.28 0.36 0.36 0.4 0.68]                   25
      [ 5  9  9 10 13 13] [0.2 0.36 0.36 0.4 0.52 0.52]                   25
      [ 5  9  9 11 11 14] [0.2 0.36 0.36 0.44 0.44 0.56]                  25
      [ 6  6  6  6  9 20] [0.24 0.24 0.24 0.24 0.36 0.8]                  25
      [ 6  6  6  6 15 16] [0.24 0.24 0.24 0.24 0.6 0.64]                  25
      [ 6  6  6  7 12 18] [0.24 0.24 0.24 0.28 0.48 0.72]                 25
      [ 6  6  8  8  8 19] [0.24 0.24 0.32 0.32 0.32 0.76]                 25
      [ 6  6  8  8 13 16] [0.24 0.24 0.32 0.32 0.52 0.64]                 25
      [ 6  6  8 10 10 17] [0.24 0.24 0.32 0.4 0.4 0.68]                   25
      [ 6  6 11 12 12 12] [0.24 0.24 0.44 0.48 0.48 0.48]                 25
      [ 6  7  7  7  9 19] [0.24 0.28 0.28 0.28 0.36 0.76]                 25
      [ 6  7  7  9 11 17] [0.24 0.28 0.28 0.36 0.44 0.68]                 25
      [ 6  7  9 11 13 13] [0.24 0.28 0.36 0.44 0.52 0.52]                 25
      [ 6  7 10 10 12 14] [0.24 0.28 0.4 0.4 0.48 0.56]                   25
      [ 6  8  8 11 12 14] [0.24 0.32 0.32 0.44 0.48 0.56]                 25
      [ 6  8 10 10 10 15] [0.24 0.32 0.4 0.4 0.4 0.6]                     25
      [ 6  9  9  9 11 15] [0.24 0.36 0.36 0.36 0.44 0.6]                  25
      [ 7  7  9  9 13 14] [0.28 0.28 0.36 0.36 0.52 0.56]                 25
      [ 7  7  9 10 11 15] [0.28 0.28 0.36 0.4 0.44 0.6]                   25
      [ 8  8  8  8 12 15] [0.32 0.32 0.32 0.32 0.48 0.6]                  25
      [ 9  9 10 11 11 11] [0.36 0.36 0.4 0.44 0.44 0.44]                  25
      [ 9 10 10 10 10 12] [0.36 0.4 0.4 0.4 0.4 0.48]                     25
      [ 1  1  3 17 18 20] [0.03125 0.03125 0.09375 0.53125 0.5625 0.625]  32
      [ 1  1  6 15 19 20] [0.03125 0.03125 0.1875 0.46875 0.59375 0.625]  32
      [ 1  1  9 16 18 19] [0.03125 0.03125 0.28125 0.5 0.5625 0.59375]    32
      [ 1  2 12 15 17 19] [0.03125 0.0625 0.375 0.46875 0.53125 0.59375]  32
      [ 1  2 13 15 15 20] [0.03125 0.0625 0.40625 0.46875 0.46875 0.625]  32
      [ 1  3  6 16 19 19] [0.03125 0.09375 0.1875 0.5 0.59375 0.59375]    32
      [ 1  3  6 17 17 20] [0.03125 0.09375 0.1875 0.53125 0.53125 0.625]  32
      [ 1  3 10 15 17 20] [0.03125 0.09375 0.3125 0.46875 0.53125 0.625]  32
      [ 1  3 11 13 18 20] [0.03125 0.09375 0.34375 0.40625 0.5625 0.625]  32
      [ 1  4 13 15 17 18] [0.03125 0.125 0.40625 0.46875 0.53125 0.5625]  32
      [ 1  4 14 15 15 19] [0.03125 0.125 0.4375 0.46875 0.46875 0.59375]  32
      [ 1  5  7 15 18 20] [0.03125 0.15625 0.21875 0.46875 0.5625 0.625]  32
      [ 1  5 12 13 18 19] [0.03125 0.15625 0.375 0.40625 0.5625 0.59375]  32
      [ 1  6  9 16 17 19] [0.03125 0.1875 0.28125 0.5 0.53125 0.59375]    32
      [ 1  6 11 12 19 19] [0.03125 0.1875 0.34375 0.375 0.59375 0.59375]  32
      [ 1  7  8 15 18 19] [0.03125 0.21875 0.25 0.46875 0.5625 0.59375]   32
      [ 1  7  9 13 18 20] [0.03125 0.21875 0.28125 0.40625 0.5625 0.625]  32
      [ 1  7 13 15 16 18] [0.03125 0.21875 0.40625 0.46875 0.5 0.5625]    32
      [ 1  8 11 15 17 18] [0.03125 0.25 0.34375 0.46875 0.53125 0.5625]   32
      [ 1  9  9 10 19 20] [0.03125 0.28125 0.28125 0.3125 0.59375 0.625]  32
      [ 1  9 10 15 16 19] [0.03125 0.28125 0.3125 0.46875 0.5 0.59375]    32
      [ 1  9 11 14 15 20] [0.03125 0.28125 0.34375 0.4375 0.46875 0.625]  32
      [ 1 11 14 15 15 16] [0.03125 0.34375 0.4375 0.46875 0.46875 0.5]    32
      [ 1 12 13 14 15 17] [0.03125 0.375 0.40625 0.4375 0.46875 0.53125]  32
      [ 2  2  6 16 18 20] [0.0625 0.0625 0.1875 0.5 0.5625 0.625]         32
      [ 2  3  5 15 19 20] [0.0625 0.09375 0.15625 0.46875 0.59375 0.625]  32
      [ 2  3  8 15 19 19] [0.0625 0.09375 0.25 0.46875 0.59375 0.59375]   32
      [ 2  3  9 13 19 20] [0.0625 0.09375 0.28125 0.40625 0.59375 0.625]  32
      [ 2  3 12 17 17 17] [0.0625 0.09375 0.375 0.53125 0.53125 0.53125]  32
      [ 2  3 13 15 16 19] [0.0625 0.09375 0.40625 0.46875 0.5 0.59375]    32
      [ 2  4 10 16 18 18] [0.0625 0.125 0.3125 0.5 0.5625 0.5625]         32
      [ 2  5  9 15 17 20] [0.0625 0.15625 0.28125 0.46875 0.53125 0.625]  32
      [ 2  5 15 15 16 17] [0.0625 0.15625 0.46875 0.46875 0.5 0.53125]    32
      [ 2  6  8 14 18 20] [0.0625 0.1875 0.25 0.4375 0.5625 0.625]        32
      [ 2  7 11 15 15 20] [0.0625 0.21875 0.34375 0.46875 0.46875 0.625]  32
      [ 2  8  9 15 17 19] [0.0625 0.25 0.28125 0.46875 0.53125 0.59375]   32
      [ 2  9  9 13 17 20] [0.0625 0.28125 0.28125 0.40625 0.53125 0.625]  32
      [ 2  9 13 15 16 17] [0.0625 0.28125 0.40625 0.46875 0.5 0.53125]    32
      [ 2 10 12 14 16 18] [0.0625 0.3125 0.375 0.4375 0.5 0.5625]         32
      [ 2 11 12 13 15 19] [0.0625 0.34375 0.375 0.40625 0.46875 0.59375]  32
      [ 3  3  7 14 19 20] [0.09375 0.09375 0.21875 0.4375 0.59375 0.625]  32
      [ 3  3 10 16 17 19] [0.09375 0.09375 0.3125 0.5 0.53125 0.59375]    32
      [ 3  3 11 14 17 20] [0.09375 0.09375 0.34375 0.4375 0.53125 0.625]  32
      [ 3  4  5 17 18 19] [0.09375 0.125 0.15625 0.53125 0.5625 0.59375]  32
      [ 3  4  9 14 19 19] [0.09375 0.125 0.28125 0.4375 0.59375 0.59375]  32
      [ 3  4 14 15 17 17] [0.09375 0.125 0.4375 0.46875 0.53125 0.53125]  32
      [ 3  4 15 15 15 18] [0.09375 0.125 0.46875 0.46875 0.46875 0.5625]  32
      [ 3  5  7 16 18 19] [0.09375 0.15625 0.21875 0.5 0.5625 0.59375]    32
      [ 3  5 11 16 17 18] [0.09375 0.15625 0.34375 0.5 0.53125 0.5625]    32
      [ 3  5 12 14 17 19] [0.09375 0.15625 0.375 0.4375 0.53125 0.59375]  32
      [ 3  5 13 14 15 20] [0.09375 0.15625 0.40625 0.4375 0.46875 0.625]  32
      [ 3  6  7 13 19 20] [0.09375 0.1875 0.21875 0.40625 0.59375 0.625]  32
      [ 3  6 11 13 17 20] [0.09375 0.1875 0.34375 0.40625 0.53125 0.625]  32
      [ 3  7  8 17 17 18] [0.09375 0.21875 0.25 0.53125 0.53125 0.5625]   32
      [ 3  7  9 14 17 20] [0.09375 0.21875 0.28125 0.4375 0.53125 0.625]  32
      [ 3  7 10 12 19 19] [0.09375 0.21875 0.3125 0.375 0.59375 0.59375]  32
      [ 3  7 11 11 18 20] [0.09375 0.21875 0.34375 0.34375 0.5625 0.625]  32
      [ 3  7 14 15 16 17] [0.09375 0.21875 0.4375 0.46875 0.5 0.53125]    32
      [ 3  8 13 13 17 18] [0.09375 0.25 0.40625 0.40625 0.53125 0.5625]   32
      [ 3  8 13 14 15 19] [0.09375 0.25 0.40625 0.4375 0.46875 0.59375]   32
      [ 3  9 10 16 17 17] [0.09375 0.28125 0.3125 0.5 0.53125 0.53125]    32
      [ 3  9 11 14 16 19] [0.09375 0.28125 0.34375 0.4375 0.5 0.59375]    32
      [ 3  9 13 13 14 20] [0.09375 0.28125 0.40625 0.40625 0.4375 0.625]  32
      [ 3 10 11 12 17 19] [0.09375 0.3125 0.34375 0.375 0.53125 0.59375]  32
      [ 3 10 11 13 15 20] [0.09375 0.3125 0.34375 0.40625 0.46875 0.625]  32
      [ 3 12 14 15 15 15] [0.09375 0.375 0.4375 0.46875 0.46875 0.46875]  32
      [ 3 13 13 14 15 16] [0.09375 0.40625 0.40625 0.4375 0.46875 0.5]    32
      [ 4  5  6 15 19 19] [0.125 0.15625 0.1875 0.46875 0.59375 0.59375]  32
      [ 4  5  9 17 17 18] [0.125 0.15625 0.28125 0.53125 0.53125 0.5625]  32
      [ 4  6  9 13 19 19] [0.125 0.1875 0.28125 0.40625 0.59375 0.59375]  32
      [ 4  6 13 15 17 17] [0.125 0.1875 0.40625 0.46875 0.53125 0.53125]  32
      [ 4  6 14 14 16 18] [0.125 0.1875 0.4375 0.4375 0.5 0.5625]         32
      [ 4  7  7 15 18 19] [0.125 0.21875 0.21875 0.46875 0.5625 0.59375]  32
      [ 4  7 11 15 17 18] [0.125 0.21875 0.34375 0.46875 0.53125 0.5625]  32
      [ 4  8 10 14 18 18] [0.125 0.25 0.3125 0.4375 0.5625 0.5625]        32
      [ 4  8 12 12 16 20] [0.125 0.25 0.375 0.375 0.5 0.625]              32
      [ 4  9  9 14 17 19] [0.125 0.28125 0.28125 0.4375 0.53125 0.59375]  32
      [ 4  9 11 11 18 19] [0.125 0.28125 0.34375 0.34375 0.5625 0.59375]  32
      [ 4 10 13 15 15 17] [0.125 0.3125 0.40625 0.46875 0.46875 0.53125]  32
      [ 4 11 13 13 15 18] [0.125 0.34375 0.40625 0.40625 0.46875 0.5625]  32
      [ 5  5  5 15 18 20] [0.15625 0.15625 0.15625 0.46875 0.5625 0.625]  32
      [ 5  5  8 15 18 19] [0.15625 0.15625 0.25 0.46875 0.5625 0.59375]   32
      [ 5  5  9 13 18 20] [0.15625 0.15625 0.28125 0.40625 0.5625 0.625]  32
      [ 5  5 13 15 16 18] [0.15625 0.15625 0.40625 0.46875 0.5 0.5625]    32
      [ 5  6  7 15 17 20] [0.15625 0.1875 0.21875 0.46875 0.53125 0.625]  32
      [ 5  6  9 11 19 20] [0.15625 0.1875 0.28125 0.34375 0.59375 0.625]  32
      [ 5  6 11 15 16 19] [0.15625 0.1875 0.34375 0.46875 0.5 0.59375]    32
      [ 5  6 12 13 17 19] [0.15625 0.1875 0.375 0.40625 0.53125 0.59375]  32
      [ 5  6 13 13 15 20] [0.15625 0.1875 0.40625 0.40625 0.46875 0.625]  32
      [ 5  7  9 16 17 18] [0.15625 0.21875 0.28125 0.5 0.53125 0.5625]    32
      [ 5  7 10 15 15 20] [0.15625 0.21875 0.3125 0.46875 0.46875 0.625]  32
      [ 5  7 11 12 18 19] [0.15625 0.21875 0.34375 0.375 0.5625 0.59375]  32
      [ 5  8  9 13 18 19] [0.15625 0.25 0.28125 0.40625 0.5625 0.59375]   32
      [ 5  8 14 15 15 17] [0.15625 0.25 0.4375 0.46875 0.46875 0.53125]   32
      [ 5  9 12 14 17 17] [0.15625 0.28125 0.375 0.4375 0.53125 0.53125]  32
      [ 5  9 12 15 15 18] [0.15625 0.28125 0.375 0.46875 0.46875 0.5625]  32
      [ 5  9 13 13 16 18] [0.15625 0.28125 0.40625 0.40625 0.5 0.5625]    32
      [ 5 10 12 13 15 19] [0.15625 0.3125 0.375 0.40625 0.46875 0.59375]  32
      [ 5 11 11 12 17 18] [0.15625 0.34375 0.34375 0.375 0.53125 0.5625]  32
      [ 6  6 10 14 16 20] [0.1875 0.1875 0.3125 0.4375 0.5 0.625]         32
      [ 6  7  8 15 17 19] [0.1875 0.21875 0.25 0.46875 0.53125 0.59375]   32
      [ 6  7  9 13 17 20] [0.1875 0.21875 0.28125 0.40625 0.53125 0.625]  32
      [ 6  7 13 15 16 17] [0.1875 0.21875 0.40625 0.46875 0.5 0.53125]    32
      [ 6  8  9 11 19 19] [0.1875 0.25 0.28125 0.34375 0.59375 0.59375]   32
      [ 6  8 10 10 18 20] [0.1875 0.25 0.3125 0.3125 0.5625 0.625]        32
      [ 6  8 11 15 17 17] [0.1875 0.25 0.34375 0.46875 0.53125 0.53125]   32
      [ 6  8 13 13 15 19] [0.1875 0.25 0.40625 0.40625 0.46875 0.59375]   32
      [ 6  9 11 13 16 19] [0.1875 0.28125 0.34375 0.40625 0.5 0.59375]    32
      [ 6  9 13 13 13 20] [0.1875 0.28125 0.40625 0.40625 0.40625 0.625]  32
      [ 6 11 11 11 15 20] [0.1875 0.34375 0.34375 0.34375 0.46875 0.625]  32
      [ 6 12 13 15 15 15] [0.1875 0.375 0.40625 0.46875 0.46875 0.46875]  32
      [ 6 12 14 14 14 16] [0.1875 0.375 0.4375 0.4375 0.4375 0.5]         32
      [ 6 13 13 13 15 16] [0.1875 0.40625 0.40625 0.40625 0.46875 0.5]    32
      [ 7  7  9 11 18 20] [0.21875 0.21875 0.28125 0.34375 0.5625 0.625]  32
      [ 7  7 11 15 16 18] [0.21875 0.21875 0.34375 0.46875 0.5 0.5625]    32
      [ 7  7 12 13 17 18] [0.21875 0.21875 0.375 0.40625 0.53125 0.5625]  32
      [ 7  7 12 14 15 19] [0.21875 0.21875 0.375 0.4375 0.46875 0.59375]  32
      [ 7  8 10 15 15 19] [0.21875 0.25 0.3125 0.46875 0.46875 0.59375]   32
      [ 7  9  9 14 16 19] [0.21875 0.28125 0.28125 0.4375 0.5 0.59375]    32
      [ 7  9 10 12 17 19] [0.21875 0.28125 0.3125 0.375 0.53125 0.59375]  32
      [ 7  9 10 13 15 20] [0.21875 0.28125 0.3125 0.40625 0.46875 0.625]  32
      [ 7 10 13 15 15 16] [0.21875 0.3125 0.40625 0.46875 0.46875 0.5]    32
      [ 7 11 12 14 15 17] [0.21875 0.34375 0.375 0.4375 0.46875 0.53125]  32
      [ 7 12 13 13 13 18] [0.21875 0.375 0.40625 0.40625 0.40625 0.5625]  32
      [ 8  9 13 14 15 17] [0.25 0.28125 0.40625 0.4375 0.46875 0.53125]   32
      [ 8 10 11 15 15 17] [0.25 0.3125 0.34375 0.46875 0.46875 0.53125]   32
      [ 8 10 12 14 14 18] [0.25 0.3125 0.375 0.4375 0.4375 0.5625]        32
      [ 8 11 11 13 15 18] [0.25 0.34375 0.34375 0.40625 0.46875 0.5625]   32
      [ 9  9 11 14 16 17] [0.28125 0.28125 0.34375 0.4375 0.5 0.53125]    32
      [ 9  9 12 13 15 18] [0.28125 0.28125 0.375 0.40625 0.46875 0.5625]  32
      [ 9 10 11 12 17 17] [0.28125 0.3125 0.34375 0.375 0.53125 0.53125]  32
      [ 9 10 12 13 13 19] [0.28125 0.3125 0.375 0.40625 0.40625 0.59375]  32
      [ 9 11 11 11 16 18] [0.28125 0.34375 0.34375 0.34375 0.5 0.5625]    32
      [ 9 11 11 12 14 19] [0.28125 0.34375 0.34375 0.375 0.4375 0.59375]  32
      [10 10 10 12 16 18] [0.3125 0.3125 0.3125 0.375 0.5 0.5625]         32
      [11 12 13 13 14 15] [0.34375 0.375 0.40625 0.40625 0.4375 0.46875]  32

Other five-dimensional integer vectors with integer norms

These are all such integer vectors that have norm less than or equal to 19. These integer vectors are grouped by the 2-norms.

  [1  1  1  1  1  2]     3
  [1  1  1  1  4  4]     6
  [1  1  1  2  2  5]     6
  [1  2  2  3  3  3]     6
  [2  2  2  2  2  4]     6
  [1  1  1  1  3  6]     7
  [1  1  2  3  3  5]     7
  [2  3  3  3  3  3]     7
  [1  1  1  2  5  7]     9
  [1  1  2  5  5  5]     9
  [1  1  3  3  5  6]     9
  [1  2  2  2  2  8]     9
  [1  2  3  3  3  7]     9
  [1  4  4  4  4  4]     9
  [2  2  2  2  4  7]     9
  [2  2  4  4  4  5]     9
  [2  3  3  3  5  5]     9
  [3  3  3  3  3  6]     9
  [1  1  1  1  6  9]    11
  [1  1  1  3  3 10]    11
  [1  1  2  3  5  9]    11
  [1  1  3  5  6  7]    11
  [1  2  3  3  7  7]    11
  [1  3  5  5  5  6]    11
  [1  4  4  4  6  6]    11
  [2  2  2  3  6  8]    11
  [2  2  4  5  6  6]    11
  [2  3  3  3  3  9]    11
  [2  3  3  5  5  7]    11
  [3  3  3  3  6  7]    11
  [1  1  1  2  4 11]    12
  [1  1  1  4  5 10]    12
  [1  1  2  5  7  8]    12
  [1  1  3  4  6  9]    12
  [1  2  4  5  7  7]    12
  [1  2  5  5  5  8]    12
  [1  3  3  3  4 10]    12
  [1  3  3  5  6  8]    12
  [2  2  2  2  8  8]    12
  [2  2  2  4  4 10]    12
  [2  3  3  3  7  8]    12
  [2  3  3  4  5  9]    12
  [2  4  4  6  6  6]    12
  [2  4  5  5  5  7]    12
  [3  3  4  5  6  7]    12
  [4  4  4  4  4  8]    12
  [1  1  1  2  9  9]    13
  [1  1  1  3  6 11]    13
  [1  1  1  6  7  9]    13
  [1  1  3  3  7 10]    13
  [1  1  5  5  6  9]    13
  [1  2  3  3  5 11]    13
  [1  2  3  5  7  9]    13
  [1  3  3  5  5 10]    13
  [1  3  5  6  7  7]    13
  [1  4  4  6  6  8]    13
  [2  2  2  2  3 12]    13
  [2  2  3  4  6 10]    13
  [2  2  5  6  6  8]    13
  [2  3  3  7  7  7]    13
  [2  3  5  5  5  9]    13
  [2  4  4  4  6  9]    13
  [3  3  3  5  6  9]    13
  [3  4  6  6  6  6]    13
  [3  5  5  5  6  7]    13
  [4  4  4  6  6  7]    13
  [1  1  1  6  6 11]    14
  [1  1  2  3  9 10]    14
  [1  1  3  4  5 12]    14
  [1  1  3  6  7 10]    14
  [1  1  4  4  9  9]    14
  [1  2  2  3  3 13]    14
  [1  2  2  5  9  9]    14
  [1  2  3  5  6 11]    14
  [1  2  5  6  7  9]    14
  [1  3  3  7  8  8]    14
  [1  3  4  5  8  9]    14
  [1  3  5  5  6 10]    14
  [1  5  6  6  7  7]    14
  [2  2  2  2  6 12]    14
  [2  2  3  3  7 11]    14
  [2  2  3  7  7  9]    14
  [2  2  4  6  6 10]    14
  [2  3  3  5  7 10]    14
  [2  3  6  7  7  7]    14
  [2  5  5  5  6  9]    14
  [3  3  3  3  4 12]    14
  [3  3  4  4  5 11]    14
  [3  3  4  7  7  8]    14
  [3  3  5  5  8  8]    14
  [3  3  5  6  6  9]    14
  [3  4  4  5  7  9]    14
  [4  6  6  6  6  6]    14
  [5  5  5  6  6  7]    14
  [1  1  1  1  5 14]    15
  [1  1  1  1 10 11]    15
  [1  1  1  2  7 13]    15
  [1  1  2  5  5 13]    15
  [1  1  2  7  7 11]    15
  [1  1  3  3  3 14]    15
  [1  1  3  3  6 13]    15
  [1  1  5  6  9  9]    15
  [1  1  5  7  7 10]    15
  [1  2  2  2  4 14]    15
  [1  2  2  4 10 10]    15
  [1  2  2  6  6 12]    15
  [1  2  3  3  9 11]    15
  [1  2  3  7  9  9]    15
  [1  2  5  5  7 11]    15
  [1  3  3  5  9 10]    15
  [1  3  3  6  7 11]    15
  [1  3  6  7  7  9]    15
  [1  4  4  8  8  8]    15
  [1  4  6  6  6 10]    15
  [1  5  5  5  7 10]    15
  [2  2  2  7  8 10]    15
  [2  2  4  4  4 13]    15
  [2  2  4  4  8 11]    15
  [2  2  5  8  8  8]    15
  [2  2  6  6  8  9]    15
  [2  3  3  3  5 13]    15
  [2  3  4  4  6 12]    15
  [2  3  5  5  9  9]    15
  [2  4  4  5  8 10]    15
  [2  5  5  5  5 11]    15
  [2  5  7  7  7  7]    15
  [2  6  6  6  7  8]    15
  [3  3  3  6  9  9]    15
  [3  3  3  7  7 10]    15
  [3  3  5  5  6 11]    15
  [3  4  6  6  8  8]    15
  [3  5  5  6  7  9]    15
  [4  4  4  7  8  8]    15
  [5  5  5  5  5 10]    15
  [1  1  1  3  9 14]    17
  [1  1  1  5  6 15]    17
  [1  1  1  6  9 13]    17
  [1  1  2  3  7 15]    17
  [1  1  2  9  9 11]    17
  [1  1  3  3 10 13]    17
  [1  1  3  6 11 11]    17
  [1  1  5  9  9 10]    17
  [1  1  6  7  9 11]    17
  [1  2  2  6 10 12]    17
  [1  2  3  5  5 15]    17
  [1  2  3  5  9 13]    17
  [1  2  4  6  6 14]    17
  [1  3  3  3  6 15]    17
  [1  3  3  5  7 14]    17
  [1  3  3  7 10 11]    17
  [1  3  5  6  7 13]    17
  [1  3  6  9  9  9]    17
  [1  3  7  7  9 10]    17
  [1  4  6  6 10 10]    17
  [1  5  5  6  9 11]    17
  [1  6  6  6  6 12]    17
  [2  2  2  4  6 15]    17
  [2  2  3  8  8 12]    17
  [2  2  6  8  9 10]    17
  [2  3  3  5 11 11]    17
  [2  3  3  7  7 13]    17
  [2  3  4  4 10 12]    17
  [2  3  5  7  9 11]    17
  [2  4  5  6  8 12]    17
  [2  5  7  7  9  9]    17
  [2  6  6  7  8 10]    17
  [3  3  3  9  9 10]    17
  [3  3  5  5  5 14]    17
  [3  3  5  5 10 11]    17
  [3  4  4  4  6 14]    17
  [3  4  6  8  8 10]    17
  [3  5  5  5  6 13]    17
  [3  5  5  7  9 10]    17
  [3  5  6  7  7 11]    17
  [4  4  4  4  9 12]    17
  [4  4  4  6  6 13]    17
  [4  4  6  6  8 11]    17
  [5  6  6  8  8  8]    17
  [5  6  7  7  7  9]    17
  [6  6  6  6  8  9]    17
  [1  1  1  1  8 16]    18
  [1  1  1  2 11 14]    18
  [1  1  1  4  4 17]    18
  [1  1  1  4  7 16]    18
  [1  1  1  5 10 14]    18
  [1  1  1 10 10 11]    18
  [1  1  2  2  5 17]    18
  [1  1  2  7 10 13]    18
  [1  1  3  5 12 12]    18
  [1  1  3  6  9 14]    18
  [1  1  4  4 11 13]    18
  [1  1  4  5  5 16]    18
  [1  1  4  8 11 11]    18
  [1  1  4  9  9 12]    18
  [1  1  5  6  6 15]    18
  [1  1  5  8  8 13]    18
  [1  1  6  6  9 13]    18
  [1  2  2  3  9 15]    18
  [1  2  2  5 11 13]    18
  [1  2  3  6  7 15]    18
  [1  2  5  5 10 13]    18
  [1  2  5  7  7 14]    18
  [1  2  6  9  9 11]    18
  [1  2  7  7 10 11]    18
  [1  3  3  3 10 14]    18
  [1  3  3  4  8 15]    18
  [1  3  3  6 10 13]    18
  [1  3  5  8  9 12]    18
  [1  3  6  6 11 11]    18
  [1  4  4  7 11 11]    18
  [1  4  5  7  8 13]    18
  [1  4  8  9  9  9]    18
  [1  5  6  9  9 10]    18
  [1  5  7  7 10 10]    18
  [1  5  7  8  8 11]    18
  [1  6  6  7  9 11]    18
  [2  2  2  4 10 14]    18
  [2  2  3  3  3 17]    18
  [2  2  4 10 10 10]    18
  [2  2  5  7 11 11]    18
  [2  2  6  6 10 12]    18
  [2  2  7  7  7 13]    18
  [2  3  3  5  9 14]    18
  [2  3  3  9 10 11]    18
  [2  3  5  5  6 15]    18
  [2  3  5  6  9 13]    18
  [2  3  7  9  9 10]    18
  [2  4  4  4  4 16]    18
  [2  4  6  6  6 14]    18
  [2  5  5  5  7 14]    18
  [2  5  5  7 10 11]    18
  [2  8  8  8 8   8]    18
  [3  3  3  3 12 12]    18
  [3  3  3  4  5 16]    18
  [3  3  3  6  6 15]    18
  [3  3  3  8  8 13]    18
  [3  3  4  4  7 15]    18
  [3  3  4  5 11 12]    18
  [3  3  5  6  7 14]    18
  [3  3  5  9 10 10]    18
  [3  3  6  7 10 11]    18
  [3  3  7  7  8 12]    18
  [3  4  4  9  9 11]    18
  [3  4  5  7  9 12]    18
  [3  5  6  6  7 13]    18
  [3  5  8  8  9  9]    18
  [3  6  6  9  9  9]    18
  [3  6  7  7  9 10]    18
  [4  4  4  4  8 14]    18
  [4  4  5  5 11 11]    18
  [4  4  5  7  7 13]    18
  [4  4  7  9  9  9]    18
  [4  4  8  8  8 10]    18
  [4  5  5  5  8 13]    18
  [4  5  7  7  8 11]    18
  [4  6  6  6 10 10]    18
  [5  5  5  7 10 10]    18
  [5  5  5  8  8 11]    18
  [5  5  6  6  9 11]    18
  [6  6  6  6  6 12]    18
  [7  7  7  7  8  8]    18
  [1  1  1  3  5 18]    19
  [1  1  1  9  9 14]    19
  [1  1  2  3 11 15]    19
  [1  1  2  7  9 15]    19
  [1  1  3  5  6 17]    19
  [1  1  3  5 10 15]    19
  [1  1  3  9 10 13]    19
  [1  1  6  7  7 15]    19
  [1  1  6  9 11 11]    19
  [1  2  2  8 12 12]    19
  [1  2  3  3  7 17]    19
  [1  2  3  3 13 13]    19
  [1  2  5  5  9 15]    19
  [1  2  5  9  9 13]    19
  [1  3  3  3  3 18]    19
  [1  3  3  5 11 14]    19
  [1  3  3  6  9 15]    19
  [1  3  3 10 11 11]    19
  [1  3  5  6 11 13]    19
  [1  3  5  7  9 14]    19
  [1  3  7  9 10 11]    19
  [1  4  4  6  6 16]    19
  [1  4  6  8 10 12]    19
  [1  5  5  6  7 15]    19
  [1  5  6  7  9 13]    19
  [1  6  9  9  9  9]    19
  [1  7  7  9  9 10]    19
  [2  2  2  3  4 18]    19
  [2  2  2  3 12 14]    19
  [2  2  2  6 12 13]    19
  [2  2  3 10 10 12]    19
  [2  2  4  7 12 12]    19
  [2  2  5  6  6 16]    19
  [2  2  8  8  9 12]    19
  [2  3  3  5  5 17]    19
  [2  3  3  7 11 13]    19
  [2  3  4  6 10 14]    19
  [2  3  5  7  7 15]    19
  [2  3  5  9 11 11]    19
  [2  3  7  7  9 13]    19
  [2  4  4  6  8 15]    19
  [2  4  4  9 10 12]    19
  [2  4  6  6 10 13]    19
  [2  5  6  6  8 14]    19
  [2  5  7  9  9 11]    19
  [2  6  6  8 10 11]    19
  [2  6  7  8  8 12]    19
  [3  3  3  3  6 17]    19
  [3  3  3  3 10 15]    19
  [3  3  5  7 10 13]    19
  [3  3  7  7  7 14]    19
  [3  3  9  9  9 10]    19
  [3  4  6 10 10 10]    19
  [3  4  8  8  8 12]    19
  [3  5  5  5  9 14]    19
  [3  5  5  9 10 11]    19
  [3  5  6  7 11 11]    19
  [3  6  6  6 10 12]    19
  [3  6  7  7  7 13]    19
  [4  4  4  5 12 12]    19
  [4  4  4  6  9 14]    19
  [4  4  6  7 10 12]    19
  [4  6  8  8  9 10]    19
  [5  5  5  5  6 15]    19
  [5  5  5  6  9 13]    19
  [5  5  7  9  9 10]    19
  [5  6  6  8 10 10]    19
  [5  6  7  7  9 11]    19