A175317 a(n) = Sum_{d|n} A007955(d) where A007955(m) = product of divisors of m.

1, 3, 4, 11, 6, 42, 8, 75, 31, 108, 12, 1778, 14, 206, 234, 1099, 18, 5901, 20, 8116, 452, 498, 24, 333618, 131, 692, 760, 22166, 30, 810372, 32, 33867, 1104, 1176, 1238, 10085333, 38, 1466, 1538, 2568180, 42, 3112382, 44, 85690, 91386, 2142, 48, 255138610

Offset: 1

Author

Jaroslav Krizek, Apr 01 2010

Keywords

• nonn

Examples

			For n = 4, with b(n) = A007955(n), a(4) = b(1) + b(2) + b(4) = 1 + 2 + 8 = 11.
		

Links

• Seiichi Manyama, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A007429, A007955, A206032, A266265.

Subsequences: A008864, A181388 \ {0}.

Programs

Mathematica

a[n_] := DivisorSum[n, #^(DivisorSigma[0, #]/2) &]; Array[a, 50] (* Amiram Eldar, Oct 23 2021 *)

PARI

a(n) = sumdiv(n, d, vecprod(divisors(d))); \\ Michel Marcus, Dec 09 2014 and Oct 23 2021

Python

from math import isqrt
from sympy import divisor_count, divisors
def A175317(n): return sum(isqrt(d)**c if (c:=divisor_count(d)) & 1 else d**(c//2) for d in divisors(n,generator=True)) # Chai Wah Wu, Jun 24 2022

Formula

From Bernard Schott, Oct 26 2021: (Start)

a(1) = 1 (the only fixed point).

a(p) = p+1 for prime p only.

a(2^k) = A181388(k+1). (End)

Extensions

Corrected by Jaroslav Krizek, Apr 02 2010

Edited and more terms from Michel Marcus, Dec 09 2014