A191161 Hypersigma(n), definition 2: sum of the divisors of n plus the recursive sum of the divisors of the proper divisors.

1, 4, 5, 12, 7, 22, 9, 32, 19, 30, 13, 72, 15, 38, 37, 80, 19, 90, 21, 96, 47, 54, 25, 208, 39, 62, 65, 120, 31, 178, 33, 192, 67, 78, 65, 316, 39, 86, 77, 272, 43, 222, 45, 168, 147, 102, 49, 560, 67, 174, 97, 192, 55

Offset: 1

Author

Alonso del Arte, May 26 2011

Keywords

• nonn
• easy

Comments

In wanting to ensure the definition was not arbitrary, I initially thought that 1s had to stop the recursion. But as T. D. Noe showed me, this doesn't have to be the case: the 1s can be included in the recursion.

Links

• Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
• Jon Maiga, Computer-generated formulas for A191161, Sequence Machine.

Crossrefs

Cf. A000203, A191150, A202687, A255242, A378211 (Dirichlet inverse).

Sequences that appear in the convolution formulas: A000010, A000203, A007429, A038040, A060640, A067824, A074206, A174725, A253249, A323910, A323912, A330575.

Programs

Mathematica

hsTD[n_] := hsTD[n] = Module[{d = Divisors[n]}, Total[d] + Total[hsTD /@ Most[d]]]; Table[hsTD[n], {n, 100}] (* From T. D. Noe *)

PARI

a(n)=sumdiv(n,d,if(dCharles R Greathouse IV, Dec 20 2011

Formula

a(n) = sigma(n) + sum_{d | n, d < n} a(d). - Charles R Greathouse IV, Dec 20 2011

From Antti Karttunen, Nov 22 2024: (Start)

Following formulas were conjectured by Sequence Machine:

For n > 1, a(n) = A191150(n) + A074206(n).

a(n) = A330575(n) + A255242(n) = 2*A255242(n) + n = 2*A330575(n) - n.

a(n) = Sum_{d|n} A330575(d).

a(n) = Sum_{d|n} d*A067824(n/d).

a(n) = Sum_{d|n} A000203(d)*A074206(n/d).

a(n) = Sum_{d|n} A007429(d)*A174725(n/d).

a(n) = Sum_{d|n} A000010(d)*A253249(n/d).

a(n) = Sum_{d|n} A038040(d)*A323912(n/d).

a(n) = Sum_{d|n} A060640(d)*A323910(n/d).

(End)