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A344327 Number of divisors of n^4.

%I A344327 #27 Aug 20 2021 16:58:09
%S A344327 1,5,5,9,5,25,5,13,9,25,5,45,5,25,25,17,5,45,5,45,25,25,5,65,9,25,13,
%T A344327 45,5,125,5,21,25,25,25,81,5,25,25,65,5,125,5,45,45,25,5,85,9,45,25,
%U A344327 45,5,65,25,65,25,25,5,225,5,25,45,25,25,125,5,45,25,125,5,117,5,25,45,45,25,125,5,85,17,25
%N A344327 Number of divisors of n^4.
%H A344327 Seiichi Manyama, <a href="/A344327/b344327.txt">Table of n, a(n) for n = 1..10000</a>
%F A344327 a(n) = A000005(A000583(n)).
%F A344327 Multiplicative with a(p^e) = 4*e+1.
%F A344327 a(n) = Sum_{d|n} 4^omega(d).
%F A344327 G.f.: Sum_{k>=1} 4^omega(k) * x^k/(1 - x^k).
%F A344327 Dirichlet g.f.: zeta(s)^2 * Product_{primes p} (1 + 3/p^s). - _Vaclav Kotesovec_, May 15 2021
%F A344327 Dirichlet g.f.: zeta(s)^5 * Product_{primes p} (1 - 6/p^(2*s) + 8/p^(3*s) - 3/p^(4*s)). - _Vaclav Kotesovec_, Aug 20 2021
%t A344327 Table[DivisorSigma[0, n^4], {n, 1, 100}] (* _Amiram Eldar_, May 15 2021 *)
%o A344327 (PARI) a(n) = numdiv(n^4);
%o A344327 (PARI) a(n) = prod(k=1, #f=factor(n)[, 2], 4*f[k]+1);
%o A344327 (PARI) a(n) = sumdiv(n, d, 4^omega(d));
%o A344327 (PARI) my(N=99, x='x+O('x^N)); Vec(sum(k=1, N, 4^omega(k)*x^k/(1-x^k)))
%o A344327 (PARI) for(n=1, 100, print1(direuler(p=2, n, (1 + 3*X)/(1 - X)^2)[n], ", ")) \\ _Vaclav Kotesovec_, May 15 2021
%o A344327 (PARI) for(n=1, 100, print1(direuler(p=2, n, (1 - 6*X^2 + 8*X^3 - 3*X^4)/(1 - X)^5)[n], ", ")) \\ _Vaclav Kotesovec_, Aug 20 2021
%Y A344327 Column k=4 of A343656.
%Y A344327 Cf. A000005, A000583, A202994.
%K A344327 nonn,mult
%O A344327 1,2
%A A344327 _Seiichi Manyama_, May 15 2021