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A344328 Number of divisors of n^5.

%I A344328 #24 Aug 19 2021 12:55:47
%S A344328 1,6,6,11,6,36,6,16,11,36,6,66,6,36,36,21,6,66,6,66,36,36,6,96,11,36,
%T A344328 16,66,6,216,6,26,36,36,36,121,6,36,36,96,6,216,6,66,66,36,6,126,11,
%U A344328 66,36,66,6,96,36,96,36,36,6,396,6,36,66,31,36,216,6,66,36,216,6,176,6,36,66,66,36
%N A344328 Number of divisors of n^5.
%H A344328 Seiichi Manyama, <a href="/A344328/b344328.txt">Table of n, a(n) for n = 1..10000</a>
%F A344328 a(n) = A000005(A000584(n)).
%F A344328 Multiplicative with a(p^e) = 5*e+1.
%F A344328 a(n) = Sum_{d|n} 5^omega(d).
%F A344328 G.f.: Sum_{k>=1} 5^omega(k) * x^k/(1 - x^k).
%F A344328 Dirichlet g.f.: zeta(s)^2 * Product_{primes p} (1 + 4/p^s). - _Vaclav Kotesovec_, Aug 19 2021
%t A344328 Table[DivisorSigma[0, n^5], {n, 1, 100}] (* _Amiram Eldar_, May 15 2021 *)
%o A344328 (PARI) a(n) = numdiv(n^5);
%o A344328 (PARI) a(n) = prod(k=1, #f=factor(n)[, 2], 5*f[k]+1);
%o A344328 (PARI) a(n) = sumdiv(n, d, 5^omega(d));
%o A344328 (PARI) my(N=99, x='x+O('x^N)); Vec(sum(k=1, N, 5^omega(k)*x^k/(1-x^k)))
%o A344328 (PARI) for(n=1, 100, print1(direuler(p=2, n, (1 + 4*X)/(1 - X)^2)[n], ", ")) \\ _Vaclav Kotesovec_, Aug 19 2021
%Y A344328 Column k=5 of A343656.
%Y A344328 Cf. A000005, A000584, A082476 (5^omega(n)), A203556.
%K A344328 nonn,mult
%O A344328 1,2
%A A344328 _Seiichi Manyama_, May 15 2021