A358380 a(n) = Sum_{d|n} tau(d^5), where tau(n) = number of divisors of n, cf. A000005.
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, 49, 34, 126, 7, 343, 7, 81, 49, 49, 49, 324, 7, 49, 49, 238, 7, 343, 7, 126, 126, 49, 7, 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
Offset: 1
Programs
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Mathematica
Array[DivisorSum[#, DivisorSigma[0, #^5] &] &, 120] (* Michael De Vlieger, Dec 13 2022 *) 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 *)
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PARI
a(n) = sumdiv(n, d, numdiv(d^5));
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PARI
a(n) = sumdiv(n, d, numdiv(n*d^3));
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PARI
a(n) = sumdiv(n, d, numdiv(n^2*d));
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PARI
a(n) = sumdiv(n, d, numdiv(n^3/d));
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PARI
my(N=80, x='x+O('x^N)); Vec(sum(k=1, N, numdiv(k^5)*x^k/(1-x^k)))
Formula
a(n) = Sum_{d|n} tau(n * d^3) = Sum_{d|n} tau(n^2 * d) = Sum_{d|n} tau(n^3 / d).
G.f.: Sum_{k>=1} tau(k^5) * x^k/(1 - x^k).
Multiplicative with a(p^e) = 5*e^2/2 + 7*e/2 + 1. - Amiram Eldar, Dec 14 2022