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%I #17 Nov 28 2021 12:35:40
%S 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,
%T 64,0,124,0,16,66,68,70,87,0,76,78,76,0,164,0,96,78,92,0,72,49,90,102,
%U 112,0,72,110,100,114,116,0,234,0,124,102,20,130,244,0,144,138,236,0,132,0,148,110
%N a(n) = Sum_{d|n} sopf(d) * sopf(n/d).
%C Dirichlet convolution of A008472 with itself.
%H Antti Karttunen, <a href="/A349712/b349712.txt">Table of n, a(n) for n = 1..20000</a>
%F Dirichlet g.f.: ( zeta(s) * primezeta(s-1) )^2.
%F a(n) = Sum_{d|n} A061397(d) * A319131(n/d).
%F a(p) = 0 for p prime. - _Michael S. Branicky_, Nov 26 2021
%F a(p^k) = (k-1)*p^2 for p prime and k > 0. - _Chai Wah Wu_, Nov 28 2021
%t sopf[n_] := DivisorSum[n, # &, PrimeQ[#] &]; a[n_] := Sum[sopf[d] sopf[n/d], {d, Divisors[n]}]; Table[a[n], {n, 1, 75}]
%o (PARI) sopf(n) = vecsum(factor(n)[, 1]); \\ A008472
%o a(n) = sumdiv(n, d, sopf(d)*sopf(n/d)); \\ _Michel Marcus_, Nov 26 2021
%o (Python)
%o from sympy import divisors, factorint
%o def sopf(n): return sum(factorint(n))
%o def a(n): return sum(sopf(d)*sopf(n//d) for d in divisors(n))
%o print([a(n) for n in range(1, 76)]) # _Michael S. Branicky_, Nov 26 2021
%Y Cf. A008472, A034761, A061397, A070288, A319131, A349711.
%K nonn
%O 1,4
%A _Ilya Gutkovskiy_, Nov 26 2021
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