A349711 a(n) = Sum_{d|n} sopfr(d) * sopfr(n/d).

0, 0, 0, 4, 0, 12, 0, 16, 9, 20, 0, 44, 0, 28, 30, 40, 0, 54, 0, 68, 42, 44, 0, 104, 25, 52, 36, 92, 0, 124, 0, 80, 66, 68, 70, 147, 0, 76, 78, 152, 0, 164, 0, 140, 108, 92, 0, 200, 49, 110, 102, 164, 0, 144, 110, 200, 114, 116, 0, 298, 0, 124, 144, 140, 130, 244, 0, 212, 138, 236, 0, 300, 0, 148, 140

Offset: 1

Author

Ilya Gutkovskiy, Nov 26 2021

Keywords

• nonn
• look

Comments

Dirichlet convolution of A001414 with itself.

Links

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

Crossrefs

Cf. A000292, A001414, A034761, A318366, A349712.

Programs

Maple

b:= proc(n) option remember; add(i[1]*i[2], i=ifactors(n)[2]) end:
a:= n-> add(b(d)*b(n/d), d=numtheory[divisors](n)):
seq(a(n), n=1..75);  # Alois P. Heinz, Nov 26 2021

Mathematica

sopfr[1] = 0; sopfr[n_] := Plus @@ Times @@@ FactorInteger@n; a[n_] := Sum[sopfr[d] sopfr[n/d], {d, Divisors[n]}]; Table[a[n], {n, 1, 75}]

PARI

sopfr(n) = (n=factor(n))[, 1]~*n[, 2]; \\ A001414
a(n) = sumdiv(n, d, sopfr(d)*sopfr(n/d)); \\ Michel Marcus, Nov 26 2021

Python

from itertools import product
from sympy import factorint
def A349711(n):
    f = factorint(n)
    plist, m = list(f.keys()), sum(f[p]*p for p in f)
    return sum((lambda x: x*(m-x))(sum(d[i]*p for i, p in enumerate(plist))) for d in product(*(list(range(f[p]+1)) for p in plist))) # Chai Wah Wu, Nov 27 2021

Formula

Dirichlet g.f.: ( zeta(s) * Sum_{p prime} p/(p^s-1) )^2.

a(p^k) = (k^3-k)*p^2/6 = A000292(k-1)*p^2 for p prime. - Chai Wah Wu, Nov 28 2021