cp's OEIS Frontend

Here congruence is relative to the 48-point cuboctahedral symmetry in a fcc lattice. The symmetric rotations and reflections of the points that comprise a(n), with redundancies removed for points that lie on axis planes, gives A386315(n).

Odd n with nonprimitive points removed gives A387223.

For all n > 0, the points at the 4 90-degree rotations of [n, 0, 0] and the eight 90-degree rotations and vertical reflections of [n/2, n/2, n*sqrt(1/2)] form the 12 vertices of a cuboctahedron (shown in red in the Links example). Points that otherwise lie on an axis plane have 24-point symmetry (green), and all other points have 48-point symmetry (blue). Thus, a(n) for all n > 0 are odd multiples of 12.

The number of noncongruent points for each sphere radius n (points within or on vertices or edges of each symmetry region in the Links example) gives A387222, the number of those points that are primitive for radius n (darker colors) gives A387223 for odd sphere radii, and the total primitive count divided by 12 gives A278081 for odd sphere radii. Examples of nonprimitive points include [3, 0, 0] and [3/2, 3/2, 3*sqrt(1/2)] for a(3), which reduce to a(1) primitive points [1, 0, 0] and [1/2, 1/2, sqrt(1/2)] respectively.

Analog for the simple cubic lattice is A016725.

Most of the initial terms are (5 * 13 * 37) * p, where p is a Pythagorean prime (A002144) other than 5, 13, or 37. These terms produce walks of 8 segments that form the same bilaterally symmetrical base shape, scaled up by a factor of p, and rotated such that the axis of symmetry becomes the hypotenuse of Pythagorean triangle [x, y, p] * 6392. See Links for examples.

Other series with their own such patterns occur with a series based on (29 * 61 * 89) * p beginning at a(30) = 787205, and another based on (13 * 109 * 229) * p beginning at a(60) = 1622465. It is conjectured that there are infinitely many of such series.

The first term that is a product of 5 different Pythagorean primes, a(44) = 1185665, is also the first that produces multiple different solutions.

It is conjectured that all terms are products of 4 or more Pythagorean primes, including at least 4 different ones, though not all such products produce closed walks. It is also conjectured that all walks have an even count of hypotenuse segments and a minimum of 6 segments, and are bilaterally symmetrical when arranged in order of increasing angle.

The shortest such walk in 2 or more dimensions, by total walk length rather than diagonal segment length, gives A385525, with A385525(2) = a(1) * 8. Closed walks along diagonals in an equilateral triangular lattice gives A387031.

All observed terms are products of 4 different primes that are 1 mod 6 (A002476), though not all such products produce closed walks. It is conjectured that all terms are products of 4 or more such primes, including at least 4 different ones.

Closed walks along diagonals in a square lattice gives A386671.

Other than a(2) whose walk is comprised of 8 diagonal segments, all known terms are produced by 3- or 4-segment walks, including some with examples of both. It is conjectured that this holds true for all n >= 3.

For n = 2, the walk segments are the hypotenuses of noncongruent primitive Pythagorean triangles.

The offset is 2, because even though the graph could be defined in dimension 1 (the vertices would be the points of Z, with each point connected to its two neighbors), it would not contain any closed paths.

Removing the constraint that the diagonal segments must all have the same length gives A386251. All such walks in 2 dimensions, by diagonal segment length rather than total walk length, gives A386671.

It is provable that all such walks must be even in total length. It is also provable that 3-segment closed walks are impossible for n < 6, and conjectured that a(n) for all n >= 6 are produced by 3-segment walks.

For n = 2, the walk segments are the hypotenuses of noncongruent primitive Pythagorean triangles.

The offset is 2, because even though the graph could be defined in dimension 1 (the vertices would be the points of Z, with each point connected to its two neighbors), it would not contain any closed paths.

Adding a constraint that the diagonal segments must all have the same length gives A385525.

Manhattan and Euclidean are distance measures from the origin for v as coordinates of a point in n-dimensional space.

Integer Euclidean(v) requires that v is a Pythagorean n-tuple.

These distances are a walk from the origin by a positive integer number of unit steps in each of the n dimensions, and a return to the origin by a straight line which is also an integer number of unit steps.

When n is a square, a(n) = n + sqrt(n) from v all 1's.

Terms are odd when n is a multiple of 5, and even otherwise.

a(10) = 103 is also the smallest number whose square is the sum of the squares of at least 2 distinct primes and which itself is also a prime.

From David A. Corneth, Jul 14 2025: (Start)

If A047432(n) is a multiple of 4 then 4 cannot be one of the squares of primes in the sum. Proof: If 4 is there the sum of squares mod 4 will be 3 (mod 4) in that case. No square is 3 (mod 4). A contradiction.

If A047432(n) is a multiple of 3 then 9 cannot be one of the squares of primes in the sum. Proof: If 9 is there then the sum of squares will be 2 (mod 3) in that case. No square is 2 (mod 3). A contradiction. (End)

Only relevant for odd primes, as every positive integer is a square mod 2.

For n >= 3, {a(n) mod 105} = {68, 83}.

For n >= 4, {a(n) mod 105} = {9, 79}.