A365442 Partial sums of A365412.

3, 18, 42, 84, 126, 189, 249, 333, 426, 546, 642, 768, 882, 1068, 1200, 1368, 1539, 1749, 1965, 2175, 2361, 2616, 2820, 3156, 3378, 3678, 3918, 4212, 4536, 4908, 5244, 5580, 5874, 6339, 6651, 7029, 7359, 7863, 8295, 8715, 9114, 9594, 9978, 10566, 11046, 11604, 12024, 12528

Offset

0,1

Comments

Partial sums of the sum of the divisors of the numbers of the form 6*k + 2, 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 second wedge after n + 1 turns. The interesting fact is that for n >> 1 the geometric pattern in the second wedge of the spiral is very similar to the geometric pattern of the fourth wedge but it is different from the other wedges. Note that the six wedge spiral shows more and better geometric patterns than the four quadrants spiral.

The graph is very close to the graph of A365444 (see the Links section).

Links

Michael De Vlieger, Table of n, a(n) for n = 0..10000

OEIS Plot 2, Plot pairs of A365442 and A365444

Omar E. Pol, Plot 6. Area of the spiral in the six wedges

Formula

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

Mathematica

Accumulate[Table[DivisorSigma[1, 6*n + 2], {n, 0, 50}]] (* Amiram Eldar, Sep 08 2023 *)

Prog

(PARI) a(n) = sum(k=0, n, sigma(6*k+2)); \\ Michel Marcus, Sep 09 2023

Crossrefs

Sequences of the same family are A363161, this sequence, A383403, A365444, A383405, A365446.

Cf. A000203, A016933, A237593, A239660, A365412.

Sequence in context: A039700 A069147 A337921 * A094159 A138976 A275038

Adjacent sequences: A365439 A365440 A365441 * A365443 A365444 A365445

Keyword

nonn,easy

Author

Omar E. Pol, Sep 07 2023

Status

approved