A353198 Intersection of A000404 and A024614.

13, 37, 52, 61, 73, 97, 109, 117, 148, 157, 169, 181, 193, 208, 229, 241, 244, 277, 292, 313, 325, 333, 337, 349, 373, 388, 397, 409, 421, 433, 436, 457, 468, 481, 541, 549, 577, 592, 601, 613, 628, 637, 657, 661, 673, 676, 709, 724, 733, 757, 769, 772, 793, 829, 832, 853, 873, 877

Offset: 1

Author

Steven Lu, May 16 2022

Keywords

• nonn

Comments

A000404 lists the numbers that are the squares of Euclidean distances between two lattice points on a square grid, the segment between which is not parallel to either axis, and thus the square grid could be divided into such number of parts symmetrically, periodically and nontrivially. Similarly, A024614 lists the numbers that are the squares of Euclidean distances between two lattice points on a hexagonal grid, the segment between which is not parallel to any of the axes.

Thus this sequence lists the numbers into which both the square grid and hexagonal one could be divided.

This sequence contains all the prime numbers of the form p = 12*k + 1 and, for each such prime, all composites of the form c = j^2*p.

Examples

			The square grid (or the Gaussian integers) can be divided into 13 parts, where the k-th part consists of grid points of the form (k + 3*n + 2*m, 2*n - 3*m) where n, m are integers. Similarly the hexagonal grid can be also divided into 13 parts, where the k-th part consists of points of the form (k + 7/2*n + m, sqrt(3)/2*n + 2*sqrt(3)*m).
		

Crossrefs

Cf. A000404, A024614.

Programs

Mathematica

Select[Intersection[
  Sort[DeleteDuplicates[
    Flatten[Table[i^2 + j^2, {i, 1000}, {j, 1000}]]]],
  Sort[DeleteDuplicates[
    Flatten[Table[i^2 + i j + j^2, {i, 1000}, {j, 1000}]]]]], # <=
   10000 &]