cp's OEIS Frontend

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

%I A383403 #47 May 22 2025 20:07:09
%S A383403 4,17,41,73,113,161,217,295,367,447,551,647,771,892,1012,1140,1296,
%T A383403 1488,1640,1822,1990,2166,2406,2598,2826,3060,3276,3564,3824,4064,
%U A383403 4312,4632,4968,5240,5552,5840,6136,6539,6923,7243,7607,7943,8375,8765,9125,9573,9989,10469,10861
%N A383403 Partial sums of the sum of the divisors of the numbers of the form 6*k + 3, k >= 0.
%C A383403 Partial sums of the sum of the divisors of A016945.
%C A383403 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.
%C A383403 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.
%H A383403 Omar E. Pol, <a href="/A383403/a383403_1.png">Illustration of a(3) = 73</a>
%H A383403 Omar E. Pol, <a href="/A383403/a383403.png">Illustration of a(10) = 551</a>
%H A383403 Omar E. Pol, <a href="/A363161/a363161.jpg">Plot 6. Area of the spiral in the six wedges</a>
%F A383403 a(n) = Sum_{k=0..n} sigma(6*k+3).
%F A383403 a(n) = (11*Pi^2/24) * n^2 + O(n*log(n)). - _Amiram Eldar_, Apr 28 2025
%e A383403 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.
%t A383403 Accumulate@ Array[DivisorSigma[1, 6 # + 3] &, 55, 0]
%o A383403 (PARI) a(n) = sum(k=0, n, sigma(6*k+3));
%Y A383403 Sequences of the same family are A363161, A365442, this sequence, A365444, A383405, A365446.
%Y A383403 Cf. A000203, A016945, A237593, A239660, A274536.
%K A383403 nonn,easy
%O A383403 0,1
%A A383403 _Omar E. Pol_, Apr 27 2025