cp's OEIS Frontend

A383405 Partial sums of the sum of the divisors of the numbers of the form 6*k + 5, k >= 0.

%I A383405 #40 May 08 2025 21:14:36
%S A383405 6,18,36,60,90,138,180,228,282,342,426,498,594,678,768,888,990,1098,
%T A383405 1212,1356,1512,1644,1782,1950,2100,2292,2484,2652,2826,3006,3234,
%U A383405 3426,3624,3864,4104,4368,4620,4848,5082,5322,5664,5916,6174,6438,6708,7080,7362,7698,7992,8328,8700,9012,9330,9690,10074
%N A383405 Partial sums of the sum of the divisors of the numbers of the form 6*k + 5, k >= 0.
%C A383405 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 fifth wedge after n + 1 turns. The interesting fact is that for n >> 1 the geometric pattern in the fifth wedge of the spiral is very similar to the geometric pattern of the first wedge but it is different from the other wedges. Also the geometric pattern in the second wedge is very similar to the geometric pattern of the fourth wedge. Note that the six wedge spiral shows more and better geometric patterns than the four quadrants spiral.
%C A383405 The graph named W5 in the Plot 6 of the Links section is very close to the graph of A363161 (W1) and far from the graph of A365446 (W6).
%H A383405 OEIS Plot 2, <a href="https://oeis.org/plot2a?name1=A363161&amp;name2=A383405&amp;tform1=untransformed&amp;tform2=untransformed&amp;shift=0&amp;radiop1=matp&amp;drawlines=true">Plot pairs of A363161 and A383405</a>
%H A383405 Omar E. Pol, <a href="/A363161/a363161.jpg">Plot 6. Area of the spiral in the six wedges</a>
%F A383405 a(n) = 6*Sum_{k=0..n} A098098(k).
%F A383405 a(n) = (Pi^2/3) * n^2 + O(n*log(n)). - _Amiram Eldar_, Apr 25 2025
%t A383405 Accumulate@ Array[DivisorSigma[1, 6 # + 5] &, 55, 0] (* _Michael De Vlieger_, Apr 25 2025 *)
%o A383405 (PARI) a(n) = sum(k=0, n, sigma(6*k+5)); \\ _Michel Marcus_, Apr 25 2025
%Y A383405 Sequences of the same family are A363161, A365442, A383403, A365444, this sequence, A365446.
%Y A383405 Cf. A000203, A016969, A098098, A237593, A239660.
%K A383405 nonn,easy
%O A383405 0,1
%A A383405 _Omar E. Pol_, Apr 25 2025