A349712 a(n) = Sum_{d|n} sopf(d) * sopf(n/d).

0, 0, 0, 4, 0, 12, 0, 8, 9, 20, 0, 32, 0, 28, 30, 12, 0, 42, 0, 48, 42, 44, 0, 52, 25, 52, 18, 64, 0, 124, 0, 16, 66, 68, 70, 87, 0, 76, 78, 76, 0, 164, 0, 96, 78, 92, 0, 72, 49, 90, 102, 112, 0, 72, 110, 100, 114, 116, 0, 234, 0, 124, 102, 20, 130, 244, 0, 144, 138, 236, 0, 132, 0, 148, 110

Offset: 1

Author

Ilya Gutkovskiy, Nov 26 2021

Keywords

• nonn

Comments

Dirichlet convolution of A008472 with itself.

Links

• Antti Karttunen, Table of n, a(n) for n = 1..20000

Crossrefs

Cf. A008472, A034761, A061397, A070288, A319131, A349711.

Programs

Mathematica

sopf[n_] := DivisorSum[n, # &, PrimeQ[#] &]; a[n_] := Sum[sopf[d] sopf[n/d], {d, Divisors[n]}]; Table[a[n], {n, 1, 75}]

PARI

sopf(n) = vecsum(factor(n)[, 1]); \\ A008472
a(n) = sumdiv(n, d, sopf(d)*sopf(n/d)); \\ Michel Marcus, Nov 26 2021

Python

from sympy import divisors, factorint
def sopf(n): return sum(factorint(n))
def a(n): return sum(sopf(d)*sopf(n//d) for d in divisors(n))
print([a(n) for n in range(1, 76)]) # Michael S. Branicky, Nov 26 2021

Formula

Dirichlet g.f.: ( zeta(s) * primezeta(s-1) )^2.

a(n) = Sum_{d|n} A061397(d) * A319131(n/d).

a(p) = 0 for p prime. - Michael S. Branicky, Nov 26 2021

a(p^k) = (k-1)*p^2 for p prime and k > 0. - Chai Wah Wu, Nov 28 2021