OFFSET
1,4
COMMENTS
Dirichlet convolution of A001414 with itself.
LINKS
Antti Karttunen, Table of n, a(n) for n = 1..20000
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
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}]
PROG
(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
CROSSREFS
KEYWORD
nonn,look
AUTHOR
Ilya Gutkovskiy, Nov 26 2021
STATUS
approved