A383403 Partial sums of the sum of the divisors of the numbers of the form 6*k + 3, k >= 0.

4, 17, 41, 73, 113, 161, 217, 295, 367, 447, 551, 647, 771, 892, 1012, 1140, 1296, 1488, 1640, 1822, 1990, 2166, 2406, 2598, 2826, 3060, 3276, 3564, 3824, 4064, 4312, 4632, 4968, 5240, 5552, 5840, 6136, 6539, 6923, 7243, 7607, 7943, 8375, 8765, 9125, 9573, 9989, 10469, 10861

Offset

0,1

Comments

Partial sums of the sum of the divisors of A016945.

See the illustration of a(3) and a(10) as the total area (or total number of cells) in the diagram of the symmetric representation of sigma in the Links section.

Also consider a spiral similar to the spiral described in A239660 but with 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 third wedge after n + 1 turns. The spiral can be visualized from the top view of the stepped pyramid described in A274536. The graph is named W3 in the Plot 6 of the Links section.

Links

Table of n, a(n) for n=0..48.

Omar E. Pol, Illustration of a(3) = 73

Omar E. Pol, Illustration of a(10) = 551

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

Formula

a(n) = Sum_{k=0..n} sigma(6*k+3).

a(n) = (11*Pi^2/24) * n^2 + O(n*log(n)). - Amiram Eldar, Apr 28 2025

Example

For n = 3 the first four terms of the numbers of the form 6*k + 3, k >= 0, are [3, 9, 15, 21]. The divisors of them are [1, 3], [1, 3, 9], [1, 3, 5, 15], [1, 3, 7, 21]. The sum of the divisors of them are [4, 13, 24, 32] respectively, and the sum of all divisors of them are 4 + 13 + 24 + 32 = 73, so a(3) = 73.

Mathematica

Accumulate@ Array[DivisorSigma[1, 6 # + 3] &, 55, 0]

Prog

(PARI) a(n) = sum(k=0, n, sigma(6*k+3));

Crossrefs

Sequences of the same family are A363161, A365442, this sequence, A365444, A383405, A365446.

Cf. A000203, A016945, A237593, A239660, A274536.

Sequence in context: A145995 A275815 A018973 * A212575 A162148 A166781

Adjacent sequences: A383400 A383401 A383402 * A383404 A383405 A383406

Keyword

nonn,easy

Author

Omar E. Pol, Apr 27 2025

Status

approved