A363161 Partial sums of A363031.

1, 9, 23, 43, 74, 106, 144, 188, 245, 317, 379, 447, 521, 601, 709, 821, 919, 1023, 1133, 1277, 1410, 1538, 1698, 1838, 2018, 2170, 2328, 2492, 2675, 2923, 3105, 3321, 3515, 3715, 3967, 4179, 4435, 4659, 4889, 5177, 5419, 5699, 5987, 6291, 6615, 6887, 7165, 7449, 7756, 8116, 8468, 8776, 9090, 9450, 9884

Offset: 0

Author

Omar E. Pol, May 18 2023

Keywords

• nonn
• easy

Comments

Partial sums of the sum of the divisors of the numbers of the form 6*k + 1, k >= 0.

From Omar E. Pol, Sep 26 2023: (Start)

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 first wedge after n turns. The interesting fact is that for n >> 1 the geometric pattern in the first wedge is similar to the geometric pattern of the fifth wedge but it is different from the other wedges. (End)

Links

• Michael De Vlieger, Table of n, a(n) for n = 0..10000
• Omar E. Pol, Plot 6. Area of the spiral in the six wedges after n turns

Crossrefs

Other sequences of the same family are A365442, A365444, A365446.

Cf. A016921, A239660, A363031.

Programs

Mathematica

Accumulate@ Array[DivisorSigma[1, 6 # + 1] &, 55, 0] (* Michael De Vlieger, Aug 27 2023 *)

PARI

a(n) = sum(k=0, n, sigma(6*k+1)); \\ Michel Marcus, Aug 28 2023

Formula

a(n) ~ Pi^2 * n^2 / 3. - Vaclav Kotesovec, Aug 31 2023