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A359037 a(n) = Sum_{d|n} tau(d^6), where tau(n) = number of divisors of n, cf. A000005.

%I A359037 #20 Dec 15 2022 09:59:44
%S A359037 1,8,8,21,8,64,8,40,21,64,8,168,8,64,64,65,8,168,8,168,64,64,8,320,21,
%T A359037 64,40,168,8,512,8,96,64,64,64,441,8,64,64,320,8,512,8,168,168,64,8,
%U A359037 520,21,168,64,168,8,320,64,320,64,64,8,1344,8,64,168,133,64,512,8,168,64,512,8,840,8
%N A359037 a(n) = Sum_{d|n} tau(d^6), where tau(n) = number of divisors of n, cf. A000005.
%H A359037 Seiichi Manyama, <a href="/A359037/b359037.txt">Table of n, a(n) for n = 1..10000</a>
%F A359037 a(n) = Sum_{d|n} tau(n * d^4) = Sum_{d|n} tau(n^2 * d^2) = Sum_{d|n} tau(n^3).
%F A359037 a(n) = tau(n) * tau(n^3).
%F A359037 G.f.: Sum_{k>=1} tau(k^6) * x^k/(1 - x^k).
%F A359037 Multiplicative with a(p^e) = 3*e^2 + 4*e + 1. - _Amiram Eldar_, Dec 14 2022
%t A359037 Array[DivisorSum[#, DivisorSigma[0, #^6] &] &, 120] (* _Michael De Vlieger_, Dec 13 2022 *)
%t A359037 f[p_, e_] := 3*e^2 + 4*e + 1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* _Amiram Eldar_, Dec 14 2022 *)
%o A359037 (PARI) a(n) = sumdiv(n, d, numdiv(d^6));
%o A359037 (PARI) a(n) = sumdiv(n, d, numdiv(n*d^4));
%o A359037 (PARI) a(n) = sumdiv(n, d, numdiv(n^2*d^2));
%o A359037 (PARI) a(n) = sumdiv(n, d, numdiv(n^3));
%o A359037 (PARI) a(n) = numdiv(n)*numdiv(n^3);
%o A359037 (PARI) my(N=80, x='x+O('x^N)); Vec(sum(k=1, N, numdiv(k^6)*x^k/(1-x^k)))
%Y A359037 Cf. A007425, A035116, A061391, A356574, A358380, A359038.
%Y A359037 Cf. A000005, A321348.
%K A359037 nonn,mult,easy
%O A359037 1,2
%A A359037 _Seiichi Manyama_, Dec 13 2022