A322671 a(n) = Sum_{d|n} (pod(d)/d), where pod(k) is the product of the divisors of k (A007955).

1, 2, 2, 4, 2, 9, 2, 12, 5, 13, 2, 155, 2, 17, 18, 76, 2, 336, 2, 415, 24, 25, 2, 13987, 7, 29, 32, 803, 2, 27035, 2, 1100, 36, 37, 38, 280418, 2, 41, 42, 64423, 2, 74133, 2, 1963, 2046, 49, 2, 5322467, 9, 2518, 54, 2735, 2, 157827, 58, 176427, 60, 61, 2

Offset: 1

Author

Jaroslav Krizek, Dec 23 2018

Keywords

• nonn

Examples

			For n = 6; a(6) = pod(1)/1 + pod(2)/2 + pod(3)/3 + pod(6)/6 = 1/1 + 2/2 + 3/3 + 36/6 = 9.
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..16384

Crossrefs

Cf. A007955, A322672.

Programs

Magma

[&+[&*[c: c in Divisors(d)] / d: d in Divisors(n)]: n in [1..100]];

Maple

pod:= proc(n) convert(numtheory:-divisors(n),`*`) end proc:
f:= proc(n) local d; add(pod(d)/d, d = numtheory:-divisors(n)) end proc:
map(f, [$1..100]); # Robert Israel, Dec 23 2018

Mathematica

Array[Sum[Apply[Times, Divisors@ d]/d, {d, Divisors@ #}] &, 59] (* Michael De Vlieger, Jan 19 2019 *)

PARI

a(n) = sumdiv(n, d, vecprod(divisors(d))/d); \\ Michel Marcus, Dec 23 2018

Python

from math import isqrt
from sympy import divisor_count, divisors
def A322671(n): return sum(isqrt(d)**(c-2) if (c:=divisor_count(d)) & 1 else d**(c//2-1) for d in divisors(n,generator=True)) # Chai Wah Wu, Jun 25 2022

Formula

a(n) = n for n = 1, 2 and 4.

a(n) = n + (tau(n) - 1) = n + 3 for squarefree semiprimes (A006881).

a(n) = 2 if n is prime. - Robert Israel, Dec 23 2018