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A358380 a(n) = Sum_{d|n} tau(d^5), where tau(n) = number of divisors of n, cf. A000005.

%I A358380 #33 Dec 14 2022 09:08:45
%S A358380 1,7,7,18,7,49,7,34,18,49,7,126,7,49,49,55,7,126,7,126,49,49,7,238,18,
%T A358380 49,34,126,7,343,7,81,49,49,49,324,7,49,49,238,7,343,7,126,126,49,7,
%U A358380 385,18,126,49,126,7,238,49,238,49,49,7,882,7,49,126,112,49,343,7,126,49,343,7,612,7,49,126,126
%N A358380 a(n) = Sum_{d|n} tau(d^5), where tau(n) = number of divisors of n, cf. A000005.
%F A358380 a(n) = Sum_{d|n} tau(n * d^3) = Sum_{d|n} tau(n^2 * d) = Sum_{d|n} tau(n^3 / d).
%F A358380 G.f.: Sum_{k>=1} tau(k^5) * x^k/(1 - x^k).
%F A358380 Multiplicative with a(p^e) = 5*e^2/2 + 7*e/2 + 1. - _Amiram Eldar_, Dec 14 2022
%t A358380 Array[DivisorSum[#, DivisorSigma[0, #^5] &] &, 120] (* _Michael De Vlieger_, Dec 13 2022 *)
%t A358380 f[p_, e_] := 5*e^2/2 + 7*e/2 + 1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* _Amiram Eldar_, Dec 14 2022 *)
%o A358380 (PARI) a(n) = sumdiv(n, d, numdiv(d^5));
%o A358380 (PARI) a(n) = sumdiv(n, d, numdiv(n*d^3));
%o A358380 (PARI) a(n) = sumdiv(n, d, numdiv(n^2*d));
%o A358380 (PARI) a(n) = sumdiv(n, d, numdiv(n^3/d));
%o A358380 (PARI) my(N=80, x='x+O('x^N)); Vec(sum(k=1, N, numdiv(k^5)*x^k/(1-x^k)))
%Y A358380 Cf. A000005, A007425, A035116, A061391, A356574, A359037, A359038.
%Y A358380 Cf. A321348.
%K A358380 nonn,mult,easy
%O A358380 1,2
%A A358380 _Seiichi Manyama_, Dec 13 2022