A191150 Hypersigma(n), definition 1: sum of the divisors of n plus the recursive sum of the divisors of the restricted divisors.

1, 3, 4, 10, 6, 19, 8, 28, 17, 27, 12, 64, 14, 35, 34, 72, 18, 82, 20, 88, 44, 51, 24, 188, 37, 59, 61, 112, 30, 165, 32, 176, 64, 75, 62, 290, 38, 83, 74, 252, 42, 209, 44, 160, 139, 99, 48, 512, 65, 166, 94, 184, 54, 306, 90, 316, 104, 123, 60, 588, 62, 131

Offset: 1

Author

Alonso del Arte, May 26 2011

Keywords

• nonn
• easy

Comments

First we add up all the divisors of n, and then we add in the divisors of each restricted divisor of n (not 1 or n itself) and continue the recursion until such a depth as that there only numbers with no restricted divisors (prime numbers).

Thus if n is prime then hypersigma(n) is the same as sigma(n).

Examples

			a(12) = 64 since: the sum of the divisors of 12 is 28; to 28 we add 3 and 4 (corresponding to the prime divisors 2 and 3) bringing us up to 35; for 4 and 6 we continue the recursion, with 4 bringing us up to 45 and 6 brings up to 64.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..10000 (first 1000 terms from Alonso del Arte)
• Jon Maiga, Computer-generated formulas for A191150, Sequence Machine.

Crossrefs

Cf. A000040, A000203, A008864, A074206, A378210 (Dirichlet inverse).

Cf. also A191161 (variant 2 of hypersigma).

Programs

Maple

a:= proc(n) option remember; uses numtheory;
      sigma(n)+add(a(d), d=divisors(n) minus {1,n})
    end:
seq(a(n), n=1..100);  # Alois P. Heinz, Aug 01 2023

Mathematica

hyperSigma[1] := 1; hyperSigma[n_] := hyperSigma[n] = Module[{d=Divisors[n]}, Total[d] + Total[hyperSigma /@ Rest[Most[d]]]]; Table[hyperSigma[n], {n, 100}] (* From T. D. Noe with a slight modification *)

PARI

A191150(n) = (sigma(n)+sumdiv(n,d,if((d>1)&&(dA191150(d), 0))); \\ (after the Maple-program) - Antti Karttunen, Nov 22 2024

Formula

a(n) = n + 1 <=> n is prime. - Bill McEachen, Aug 01 2023

For n > 1, a(n) = A191161(n) - A074206(n). [Conjectured by Sequence Machine] - Antti Karttunen, Nov 22 2024

Extensions

Example corrected by Paolo P. Lava, Jul 13 2011