A349711 - OEIS

%I #22 Nov 28 2021 12:09:32

%S 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,

%T 36,92,0,124,0,80,66,68,70,147,0,76,78,152,0,164,0,140,108,92,0,200,

%U 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

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

%C Dirichlet convolution of A001414 with itself.

%H Antti Karttunen, <a href="/A349711/b349711.txt">Table of n, a(n) for n = 1..20000</a>

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

%F a(p^k) = (k^3-k)*p^2/6 = A000292(k-1)*p^2 for p prime. - _Chai Wah Wu_, Nov 28 2021

%p b:= proc(n) option remember; add(i[1]*i[2], i=ifactors(n)[2]) end:

%p a:= n-> add(b(d)*b(n/d), d=numtheory[divisors](n)):

%p seq(a(n), n=1..75);  # _Alois P. Heinz_, Nov 26 2021

%t 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}]

%o (PARI) sopfr(n) = (n=factor(n))[, 1]~*n[, 2]; \\ A001414

%o a(n) = sumdiv(n, d, sopfr(d)*sopfr(n/d)); \\ _Michel Marcus_, Nov 26 2021

%o (Python)

%o from itertools import product

%o from sympy import factorint

%o def A349711(n):

%o     f = factorint(n)

%o     plist, m = list(f.keys()), sum(f[p]*p for p in f)

%o     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

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

%K nonn,look

%O 1,4

%A _Ilya Gutkovskiy_, Nov 26 2021