A365444 Partial sums of A365414.

7, 25, 56, 92, 148, 202, 292, 364, 462, 552, 679, 823, 963, 1089, 1269, 1413, 1630, 1792, 2040, 2220, 2444, 2696, 2966, 3182, 3448, 3736, 4114, 4366, 4674, 4944, 5304, 5664, 6063, 6369, 6803, 7127, 7631, 7973, 8423, 8855, 9289, 9757, 10268, 10664, 11140, 11554, 12274, 12778

Offset: 0

Author

Omar E. Pol, Sep 07 2023

Keywords

• nonn
• easy
• less

Comments

Partial sums of the sum of the divisors of the numbers of the form 6*k + 4, k >= 0.

Consider a spiral similar to the spiral described in A239660 but instead of having four quadrants on the square grid the new spiral has six wedges on the triangular grid. A "diamond" formed by two adjacent triangles has area 1. a(n) is the total number of diamonds (or the total area) in the fourth wedge after n turns. The interesting fact is that for n >> 1 the geometric pattern in the fourth wedge of the spiral is similar to the geometric pattern of the second wedge but it is different from the other wedges.

Links

• OEIS Plot 2, Plot pairs of A365444 and A365442.
• Omar E. Pol, Plot 6. Area of the spiral in the six wedges after n turns

Crossrefs

Other sequences of the same family are A363161, A365442, A365446.

Cf. A000203, A016957, A239660, A365414.

Programs

Mathematica

Accumulate[Table[DivisorSigma[1, 6*n + 4], {n, 0, 50}]] (* Amiram Eldar, Sep 08 2023 *)

PARI

a(n) = sum(k=0, n, sigma(6*k+4)); \\ Michel Marcus, Sep 08 2023

Formula

a(n) = (5*Pi^2/9) * n^2 + O(n*log(n)). - Amiram Eldar, Sep 08 2023