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A359038 a(n) = Sum_{d|n} tau(d^7), where tau(n) = number of divisors of n, cf. A000005.

%I A359038 #24 Dec 14 2022 09:08:36
%S A359038 1,9,9,24,9,81,9,46,24,81,9,216,9,81,81,75,9,216,9,216,81,81,9,414,24,
%T A359038 81,46,216,9,729,9,111,81,81,81,576,9,81,81,414,9,729,9,216,216,81,9,
%U A359038 675,24,216,81,216,9,414,81,414,81,81,9,1944,9,81,216,154,81,729,9,216,81,729,9
%N A359038 a(n) = Sum_{d|n} tau(d^7), where tau(n) = number of divisors of n, cf. A000005.
%F A359038 a(n) = Sum_{d|n} tau(n * d^5) = Sum_{d|n} tau(n^2 * d^3) = Sum_{d|n} tau(n^3 * d) = Sum_{d|n} tau(n^4 / d).
%F A359038 G.f.: Sum_{k>=1} tau(k^7) * x^k/(1 - x^k).
%F A359038 Multiplicative with a(p^e) = 7*e^2/2 + 9*e/2 + 1. - _Amiram Eldar_, Dec 14 2022
%t A359038 Array[DivisorSum[#, DivisorSigma[0, #^7] &] &, 120] (* _Michael De Vlieger_, Dec 13 2022 *)
%t A359038 f[p_, e_] := 7*e^2/2 + 9*e/2 + 1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* _Amiram Eldar_, Dec 14 2022 *)
%o A359038 (PARI) a(n) = sumdiv(n, d, numdiv(d^7));
%o A359038 (PARI) a(n) = sumdiv(n, d, numdiv(n*d^5));
%o A359038 (PARI) a(n) = sumdiv(n, d, numdiv(n^2*d^3));
%o A359038 (PARI) a(n) = sumdiv(n, d, numdiv(n^3*d));
%o A359038 (PARI) a(n) = sumdiv(n, d, numdiv(n^4/d));
%o A359038 (PARI) my(N=80, x='x+O('x^N)); Vec(sum(k=1, N, numdiv(k^7)*x^k/(1-x^k)))
%o A359038 (Python)
%o A359038 from math import prod
%o A359038 from sympy import factorint
%o A359038 def A359038(n): return prod((e+1)*(7*e+2)>>1 for e in factorint(n).values()) # _Chai Wah Wu_, Dec 13 2022
%Y A359038 Cf. A000005, A007425, A035116, A061391, A356574, A358380, A359037.
%Y A359038 Cf. A321348.
%K A359038 nonn,mult,easy
%O A359038 1,2
%A A359038 _Seiichi Manyama_, Dec 13 2022