A365446 Partial sums of A224613.

12, 40, 79, 139, 211, 302, 398, 522, 642, 810, 954, 1149, 1317, 1541, 1775, 2027, 2243, 2523, 2763, 3123, 3435, 3771, 4059, 4462, 4834, 5226, 5589, 6069, 6429, 6975, 7359, 7867, 8335, 8839, 9415, 10015, 10471, 11031, 11577, 12321, 12825, 13553, 14081, 14801, 15521, 16193, 16769, 17588, 18272, 19140, 19842

Offset: 1

Author

Omar E. Pol, Sep 04 2023

Keywords

• nonn
• easy

Comments

Partial sums of the sum of the divisors of the nonzero multiples of 6.

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 sixth wedge after n turns. Note that the six wedge spiral shows more and better geometric patterns than the four quadrants spiral. - Omar E. Pol, Apr 26 2025

Links

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

Crossrefs

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

Cf. A000203, A008588, A224613, A237593, A239660.

Programs

Mathematica

Accumulate[Table[DivisorSigma[1, 6*n], {n, 1, 50}]] (* Amiram Eldar, Sep 07 2023 *)

Python

from math import prod
from collections import Counter
from sympy import factorint
def A365446(n): return sum(prod((p**(e+1)-1)//(p-1) for p, e in (Counter(factorint(m))+Counter([2,3])).items()) for m in range(1,n+1)) # Chai Wah Wu, Sep 07 2023

Formula

a(n) = (55*Pi^2/72) * n^2 + O(n*log(n)). - Amiram Eldar, Sep 07 2023