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

1,2

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.

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)

Mathematica

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

Prog

(PARI) a(n)=sumdiv(n, d, if(d<n, d+a(d), n)) \\ Charles R Greathouse IV, Dec 20 2011

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.

Sequence in context: A260624 A067371 A068719 * A246316 A344372 A034773

Adjacent sequences: A191158 A191159 A191160 * A191162 A191163 A191164

Keyword

nonn,easy

Author

Alonso del Arte, May 26 2011

Status

approved