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%I #63 Dec 15 2022 09:59:49
%S 1,6,6,15,6,36,6,28,15,36,6,90,6,36,36,45,6,90,6,90,36,36,6,168,15,36,
%T 28,90,6,216,6,66,36,36,36,225,6,36,36,168,6,216,6,90,90,36,6,270,15,
%U 90,36,90,6,168,36,168,36,36,6,540,6,36,90,91,36,216,6,90,36,216,6,420,6,36,90,90
%N a(n) = Sum_{d|n} tau(d^4), where tau(n) = number of divisors of n, cf. A000005.
%H Seiichi Manyama, <a href="/A356574/b356574.txt">Table of n, a(n) for n = 1..10000</a>
%F a(n) = Sum_{d|n} tau(n * d^2) = Sum_{d|n} tau(n^2).
%F a(n) = tau(n) * tau(n^2).
%F G.f.: Sum_{k>=1} tau(k^4) * x^k/(1 - x^k).
%F Multiplicative with a(p^e) = 2*e^2 + 3*e + 1. - _Amiram Eldar_, Dec 14 2022
%t Array[DivisorSum[#, DivisorSigma[0, #^4] &] &, 120] (* _Michael De Vlieger_, Dec 13 2022 *)
%t f[p_, e_] := 2*e^2 + 3*e + 1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* _Amiram Eldar_, Dec 14 2022 *)
%o (PARI) a(n) = sumdiv(n, d, numdiv(d^4));
%o (PARI) a(n) = sumdiv(n, d, numdiv(n*d^2));
%o (PARI) a(n) = sumdiv(n, d, numdiv(n^2));
%o (PARI) a(n) = numdiv(n)*numdiv(n^2);
%o (PARI) my(N=80, x='x+O('x^N)); Vec(sum(k=1, N, numdiv(k^4)*x^k/(1-x^k)))
%o (Python)
%o from math import prod
%o from sympy import factorint
%o def A356574(n): return prod((e+1)*((e<<1)+1) for e in factorint(n).values()) # _Chai Wah Wu_, Dec 13 2022
%Y Cf. A007425, A035116, A061391, A358380, A359037, A359038.
%Y Cf. A000005, A321348.
%K nonn,easy,mult
%O 1,2
%A _Seiichi Manyama_, Dec 13 2022
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