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Luis Mendo
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A positive integer \$k\$ is a Loeschian number if

  • \$k\$ can be expressed as \$i^2 + j^2 + i\times j\$ for \$i\$, \$j\$ integers.

For example, the first positive Loeschian numbers are: \$1\$ (\$i=1, j=0\$); \$3\$ (\$i=j=1\$); \$4\$ (\$i=2, j=0\$); \$7\$ (\$i=2, j=1\$); \$9\$ (\$i=-3, j=3\$); ... Note that \$i, j\$ for a given \$k\$ are not unique. For example, \$9\$ can also be generated with \$i=3, j=0\$.

Other equivalent characterizations of these numbers are:

  • \$k\$ can be expressed as \$i^2 + j^2 + i\times j\$ for \$i, j\$ non-negative integers. (For each pair of integers \$i, j\$ there's a pair of non-negative integers that gives the same \$k\$)

  • There is a set of \$k\$ contiguous hexagons that forms a tesselation on a hexagonal grid so that all \$k\$ corresponding cells are the same distance apart (see illustrations for \$k = 4\$ and for \$k = 7\$). (Because of this property, these numbers find application in mobile cellular communication networks.)

  • See more characterizations in the OEIS page of the sequence.

The challenge

Given a positive integer, output a truthy result if it is a Loeschian number, or a falsy result otherwise.

The program or function should handle (say in less than a minute) inputs up to \$1000\$, or up to data type limitations.

Code golf. Shortest wins.

Test cases

The following numbers should output a truthy result:

1, 4, 7, 12, 13, 108, 109, 192, 516, 999

The following numbers should output a falsy result:

2, 5, 10, 42, 101, 102, 128, 150, 501, 1000

A positive integer \$k\$ is a Loeschian number if

  • \$k\$ can be expressed as \$i^2 + j^2 + i\times j\$ for \$i\$, \$j\$ integers.

For example, the first positive Loeschian numbers are: \$1\$ (\$i=1, j=0\$); \$3\$ (\$i=j=1\$); \$4\$ (\$i=2, j=0\$); \$7\$ (\$i=2, j=1\$); \$9\$ (\$i=-3, j=3\$); ... Note that \$i, j\$ for a given \$k\$ are not unique. For example, \$9\$ can also be generated with \$i=3, j=0\$.

Other equivalent characterizations of these numbers are:

  • \$k\$ can be expressed as \$i^2 + j^2 + i\times j\$ for \$i, j\$ non-negative integers. (For each pair of integers \$i, j\$ there's a pair of non-negative integers that gives the same \$k\$)

  • There is a set of \$k\$ contiguous hexagons that forms a tesselation on a hexagonal grid so that all \$k\$ corresponding cells are the same distance apart (see illustrations for \$k = 4\$ and for \$k = 7\$). (Because of this property, these numbers find application in mobile cellular communication networks.)

  • See more characterizations in the OEIS page of the sequence.

The challenge

Given a positive integer, output a truthy result if it is a Loeschian number, or a falsy result otherwise.

The program or function should handle (say in less than a minute) inputs up to \$1000\$, or up to data type limitations.

Code golf. Shortest wins.

Test cases

The following numbers should output a truthy result:

1, 4, 7, 12, 13, 108, 109, 192, 516, 999

The following numbers should output a falsy result:

2, 5, 10, 42, 101, 102, 128, 150, 501, 1000

A positive integer \$k\$ is a Loeschian number if

  • \$k\$ can be expressed as \$i^2 + j^2 + i\times j\$ for \$i\$, \$j\$ integers.

For example, the first positive Loeschian numbers are: \$1\$ (\$i=1, j=0\$); \$3\$ (\$i=j=1\$); \$4\$ (\$i=2, j=0\$); \$7\$ (\$i=2, j=1\$); \$9\$ (\$i=-3, j=3\$); ... Note that \$i, j\$ for a given \$k\$ are not unique. For example, \$9\$ can also be generated with \$i=3, j=0\$.

Other equivalent characterizations of these numbers are:

  • \$k\$ can be expressed as \$i^2 + j^2 + i\times j\$ for \$i, j\$ non-negative integers. (For each pair of integers \$i, j\$ there's a pair of non-negative integers that gives the same \$k\$)

  • There is a set of \$k\$ contiguous hexagons that forms a tesselation on a hexagonal grid so that corresponding cells are the same distance apart (see illustrations for \$k = 4\$ and for \$k = 7\$). (Because of this property, these numbers find application in mobile cellular communication networks.)

  • See more characterizations in the OEIS page of the sequence.

The challenge

Given a positive integer, output a truthy result if it is a Loeschian number, or a falsy result otherwise.

The program or function should handle (say in less than a minute) inputs up to \$1000\$, or up to data type limitations.

Code golf. Shortest wins.

Test cases

The following numbers should output a truthy result:

1, 4, 7, 12, 13, 108, 109, 192, 516, 999

The following numbers should output a falsy result:

2, 5, 10, 42, 101, 102, 128, 150, 501, 1000
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Luis Mendo
Luis Mendo
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A positive integer \$k\$ is a Loeschian number if

  • \$k\$ can be expressed as \$i^2 + j^2 + i\times j\$ for \$i\$, \$j\$ integers.

For example, the first positive Loeschian numbers are: \$1\$ (\$i=1, j=0\$); \$3\$ (\$i=j=1\$); \$4\$ (\$i=2, j=0\$); \$7\$ (\$i=2, j=1\$); \$9\$ (\$i=-3, j=3\$); ... Note that \$i, j\$ for a given \$k\$ are not unique. For example, \$9\$ can also be generated with \$i=3, j=0\$.

Other equivalent characterizations of these numbers are:

  • \$k\$ can be expressed as \$i^2 + j^2 + i\times j\$ for \$i, j\$ non-negative integers. (For each pair of integers \$i, j\$ there's a pair of non-negative integers that gives the same \$k\$)

  • There is a set of \$k\$ contiguous hexagons that forms a tesselation on a hexagonal grid so that all \$k\$ corresponding cells are the same distance apart (see illustrations for \$k = 4\$ and for \$k = 7\$). (Because of this property, these numbers find application in mobile cellular communication networks.)

  • See more characterizations in the OEIS page of the sequence.

The challenge

Given a positive integer, output a truthy result if it is a Loeschian number, or a falsy result otherwise.

The program or function should handle (say in less than a minute) inputs up to \$1000\$, or up to data type limitations.

Code golf. Shortest wins.

Test cases

The following numbers should output a truthy result:

1, 4, 7, 12, 13, 108, 109, 192, 516, 999

The following numbers should output a falsy result:

2, 5, 10, 42, 101, 102, 128, 150, 501, 1000

A positive integer \$k\$ is a Loeschian number if

  • \$k\$ can be expressed as \$i^2 + j^2 + i\times j\$ for \$i\$, \$j\$ integers.

For example, the first positive Loeschian numbers are: \$1\$ (\$i=1, j=0\$); \$3\$ (\$i=j=1\$); \$4\$ (\$i=2, j=0\$); \$7\$ (\$i=2, j=1\$); \$9\$ (\$i=-3, j=3\$); ... Note that \$i, j\$ for a given \$k\$ are not unique. For example, \$9\$ can also be generated with \$i=3, j=0\$.

Other equivalent characterizations of these numbers are:

  • \$k\$ can be expressed as \$i^2 + j^2 + i\times j\$ for \$i, j\$ non-negative integers. (For each pair of integers \$i, j\$ there's a pair of non-negative integers that gives the same \$k\$)

  • There is a set of \$k\$ contiguous hexagons that forms a tesselation on a hexagonal grid (see illustrations for \$k = 4\$ and for \$k = 7\$). (Because of this property, these numbers find application in mobile cellular communication networks.)

  • See more characterizations in the OEIS page of the sequence.

The challenge

Given a positive integer, output a truthy result if it is a Loeschian number, or a falsy result otherwise.

The program or function should handle (say in less than a minute) inputs up to \$1000\$, or up to data type limitations.

Code golf. Shortest wins.

Test cases

The following numbers should output a truthy result:

1, 4, 7, 12, 13, 108, 109, 192, 516, 999

The following numbers should output a falsy result:

2, 5, 10, 42, 101, 102, 128, 150, 501, 1000

A positive integer \$k\$ is a Loeschian number if

  • \$k\$ can be expressed as \$i^2 + j^2 + i\times j\$ for \$i\$, \$j\$ integers.

For example, the first positive Loeschian numbers are: \$1\$ (\$i=1, j=0\$); \$3\$ (\$i=j=1\$); \$4\$ (\$i=2, j=0\$); \$7\$ (\$i=2, j=1\$); \$9\$ (\$i=-3, j=3\$); ... Note that \$i, j\$ for a given \$k\$ are not unique. For example, \$9\$ can also be generated with \$i=3, j=0\$.

Other equivalent characterizations of these numbers are:

  • \$k\$ can be expressed as \$i^2 + j^2 + i\times j\$ for \$i, j\$ non-negative integers. (For each pair of integers \$i, j\$ there's a pair of non-negative integers that gives the same \$k\$)

  • There is a set of \$k\$ contiguous hexagons that forms a tesselation on a hexagonal grid so that all \$k\$ corresponding cells are the same distance apart (see illustrations for \$k = 4\$ and for \$k = 7\$). (Because of this property, these numbers find application in mobile cellular communication networks.)

  • See more characterizations in the OEIS page of the sequence.

The challenge

Given a positive integer, output a truthy result if it is a Loeschian number, or a falsy result otherwise.

The program or function should handle (say in less than a minute) inputs up to \$1000\$, or up to data type limitations.

Code golf. Shortest wins.

Test cases

The following numbers should output a truthy result:

1, 4, 7, 12, 13, 108, 109, 192, 516, 999

The following numbers should output a falsy result:

2, 5, 10, 42, 101, 102, 128, 150, 501, 1000
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caird coinheringaahin g ♦
caird coinheringaahin g ♦
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A positive integer k\$k\$ is a Loeschian number if

  • k\$k\$ can be expressed as i*i + j*j + i*j\$i^2 + j^2 + i\times j\$ for i\$i\$, j\$j\$ integers.

For example, the first positive Loeschian numbers are: 1\$1\$ (i=1, j=0\$i=1, j=0\$); 3\$3\$ (i=j=1\$i=j=1\$); 4\$4\$ (i=2, j=0\$i=2, j=0\$); 7\$7\$ (i=2, j=1\$i=2, j=1\$); 9\$9\$ (i=-3, j=3\$i=-3, j=3\$); ... Note that i, j\$i, j\$ for a given k\$k\$ are not unique. For example, 9\$9\$ can also be generated with i=3, j=0\$i=3, j=0\$.

Other equivalent characterizations of these numbers are:

  • k\$k\$ can be expressed as i*i + j*j + i*j\$i^2 + j^2 + i\times j\$ for i, j\$i, j\$ non-negative integers. (For each pair of integers i, j\$i, j\$ there's a pair of non-negative integers that gives the same k\$k\$)

  • There is a set of k\$k\$ contiguous hexagons that forms a tesselation on a hexagonal grid (see illustrations for k = 4\$k = 4\$ and for k = 7\$k = 7\$). (Because of this property, these numbers find application in mobile cellular communication networks.)

  • See more characterizations in the OEIS page of the sequence.

The challenge

Given a positive integer, output a truthy result if it is a Loeschian number, or a falsy result otherwise.

The program or function should handle (say in less than a minute) inputs up to 1000\$1000\$, or up to data type limitations.

Code golf. Shortest wins.

Test cases

The following numbers should output a truthy result:

1, 4, 7, 12, 13, 108, 109, 192, 516, 999

The following numbers should output a falsy result:

2, 5, 10, 42, 101, 102, 128, 150, 501, 1000

A positive integer k is a Loeschian number if

  • k can be expressed as i*i + j*j + i*j for i, j integers.

For example, the first positive Loeschian numbers are: 1 (i=1, j=0); 3 (i=j=1); 4 (i=2, j=0); 7 (i=2, j=1); 9 (i=-3, j=3); ... Note that i, j for a given k are not unique. For example, 9 can also be generated with i=3, j=0.

Other equivalent characterizations of these numbers are:

  • k can be expressed as i*i + j*j + i*j for i, j non-negative integers. (For each pair of integers i, j there's a pair of non-negative integers that gives the same k)

  • There is a set of k contiguous hexagons that forms a tesselation on a hexagonal grid (see illustrations for k = 4 and for k = 7). (Because of this property, these numbers find application in mobile cellular communication networks.)

  • See more characterizations in the OEIS page of the sequence.

The challenge

Given a positive integer, output a truthy result if it is a Loeschian number, or a falsy result otherwise.

The program or function should handle (say in less than a minute) inputs up to 1000, or up to data type limitations.

Code golf. Shortest wins.

Test cases

The following numbers should output a truthy result:

1, 4, 7, 12, 13, 108, 109, 192, 516, 999

The following numbers should output a falsy result:

2, 5, 10, 42, 101, 102, 128, 150, 501, 1000

A positive integer \$k\$ is a Loeschian number if

  • \$k\$ can be expressed as \$i^2 + j^2 + i\times j\$ for \$i\$, \$j\$ integers.

For example, the first positive Loeschian numbers are: \$1\$ (\$i=1, j=0\$); \$3\$ (\$i=j=1\$); \$4\$ (\$i=2, j=0\$); \$7\$ (\$i=2, j=1\$); \$9\$ (\$i=-3, j=3\$); ... Note that \$i, j\$ for a given \$k\$ are not unique. For example, \$9\$ can also be generated with \$i=3, j=0\$.

Other equivalent characterizations of these numbers are:

  • \$k\$ can be expressed as \$i^2 + j^2 + i\times j\$ for \$i, j\$ non-negative integers. (For each pair of integers \$i, j\$ there's a pair of non-negative integers that gives the same \$k\$)

  • There is a set of \$k\$ contiguous hexagons that forms a tesselation on a hexagonal grid (see illustrations for \$k = 4\$ and for \$k = 7\$). (Because of this property, these numbers find application in mobile cellular communication networks.)

  • See more characterizations in the OEIS page of the sequence.

The challenge

Given a positive integer, output a truthy result if it is a Loeschian number, or a falsy result otherwise.

The program or function should handle (say in less than a minute) inputs up to \$1000\$, or up to data type limitations.

Code golf. Shortest wins.

Test cases

The following numbers should output a truthy result:

1, 4, 7, 12, 13, 108, 109, 192, 516, 999

The following numbers should output a falsy result:

2, 5, 10, 42, 101, 102, 128, 150, 501, 1000
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Luis Mendo
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