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

6, 18, 36, 60, 90, 138, 180, 228, 282, 342, 426, 498, 594, 678, 768, 888, 990, 1098, 1212, 1356, 1512, 1644, 1782, 1950, 2100, 2292, 2484, 2652, 2826, 3006, 3234, 3426, 3624, 3864, 4104, 4368, 4620, 4848, 5082, 5322, 5664, 5916, 6174, 6438, 6708, 7080, 7362, 7698, 7992, 8328, 8700, 9012, 9330, 9690, 10074

Offset: 0

Author

Omar E. Pol, Apr 25 2025

Keywords

• nonn
• easy

Comments

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.

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).

Links

• OEIS Plot 2, Plot pairs of A363161 and A383405
• Omar E. Pol, Plot 6. Area of the spiral in the six wedges

Crossrefs

Sequences of the same family are A363161, A365442, A383403, A365444, this sequence, A365446.

Cf. A000203, A016969, A098098, A237593, A239660.

Programs

Mathematica

Accumulate@ Array[DivisorSigma[1, 6 # + 5] &, 55, 0] (* Michael De Vlieger, Apr 25 2025 *)

PARI

a(n) = sum(k=0, n, sigma(6*k+5)); \\ Michel Marcus, Apr 25 2025

Formula

a(n) = 6*Sum_{k=0..n} A098098(k).

a(n) = (Pi^2/3) * n^2 + O(n*log(n)). - Amiram Eldar, Apr 25 2025