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Partial sums of A365414.
4

%I #34 Oct 22 2023 16:58:26

%S 7,25,56,92,148,202,292,364,462,552,679,823,963,1089,1269,1413,1630,

%T 1792,2040,2220,2444,2696,2966,3182,3448,3736,4114,4366,4674,4944,

%U 5304,5664,6063,6369,6803,7127,7631,7973,8423,8855,9289,9757,10268,10664,11140,11554,12274,12778

%N Partial sums of A365414.

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

%C 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.

%H OEIS Plot 2, <a href="https://oeis.org/plot2a?name1=A365444&amp;name2=A365442&amp;tform1=untransformed&amp;tform2=untransformed&amp;shift=0&amp;radiop1=matp&amp;drawlines=true">Plot pairs of A365444 and A365442</a>.

%H Omar E. Pol, <a href="/A365444/a365444.jpg">Plot 6. Area of the spiral in the six wedges after n turns</a>

%F a(n) = (5*Pi^2/9) * n^2 + O(n*log(n)). - _Amiram Eldar_, Sep 08 2023

%t Accumulate[Table[DivisorSigma[1, 6*n + 4], {n, 0, 50}]] (* _Amiram Eldar_, Sep 08 2023 *)

%o (PARI) a(n) = sum(k=0, n, sigma(6*k+4)); \\ _Michel Marcus_, Sep 08 2023

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

%Y Cf. A000203, A016957, A239660, A365414.

%K nonn,easy,less

%O 0,1

%A _Omar E. Pol_, Sep 07 2023