A280685 a(n) = sum of the divisors of the product of the divisors of n.

1, 3, 4, 15, 6, 91, 8, 127, 40, 217, 12, 5080, 14, 399, 403, 2047, 18, 16395, 20, 19812, 741, 931, 24, 991111, 156, 1281, 1093, 50800, 30, 2929531, 32, 65535, 1729, 2149, 1767, 30203052, 38, 2667, 2379, 6397171, 42, 10506551, 44, 185928, 170508, 3871, 48

Offset

1,2

Links

Robert Israel, Table of n, a(n) for n = 1..10000

Formula

a(n) = A000203(A007955(n)).

a(p) = p + 1 for p = primes (A000040).

a(2^n) = 2*A007955(2^n) - 1. [corrected by Jason Yuen, Mar 08 2025]

Example

For n = 4; a(n) = sigma (1*2*4) = sigma(8) = 15.

Maple

f:= proc(n) local F, q, t, E, i, j;
  F:= ifactors(n)[2];
  q:= mul(1+t, t=F[.., 2]);
  E:= [seq(q*F[i, 2]/2, i=1..nops(F))];
  mul(add(F[i, 1]^j, j=0..E[i]), i=1..nops(F))
end proc:
map(f, [$1..50]); # Robert Israel, Mar 08 2026

Mathematica

A280685[n_] := DivisorSigma[1, Times @@ Divisors[n]];
Array[A280685, 50] (* Paolo Xausa, Mar 09 2026 *)

Prog

(Magma) [&+[d: d in Divisors(&*[d: d in Divisors(n)])]: n in [1..100]];
(Python)
from math import isqrt
from sympy import divisor_sigma
def A280685(n): return divisor_sigma((isqrt(n) if (c:=divisor_count(n)) & 1 else 1)*n**(c//2)) # Chai Wah Wu, Jun 25 2022
(PARI) a(n) = sigma(vecprod(divisors(n))); \\ Michel Marcus, Mar 09 2026

Crossrefs

Cf. A000040, A000203, A007955, A280582, A280684.

Sequence in context: A194924 A205489 A343935 * A076365 A070971 A338438

Adjacent sequences: A280682 A280683 A280684 * A280686 A280687 A280688

Keyword

nonn

Author

Jaroslav Krizek, Jan 07 2017

Status

approved