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A344335 Number of divisors of n^8.

%I A344335 #16 Aug 19 2021 12:55:24
%S A344335 1,9,9,17,9,81,9,25,17,81,9,153,9,81,81,33,9,153,9,153,81,81,9,225,17,
%T A344335 81,25,153,9,729,9,41,81,81,81,289,9,81,81,225,9,729,9,153,153,81,9,
%U A344335 297,17,153,81,153,9,225,81,225,81,81,9,1377,9,81,153,49,81,729,9,153,81,729,9
%N A344335 Number of divisors of n^8.
%H A344335 Seiichi Manyama, <a href="/A344335/b344335.txt">Table of n, a(n) for n = 1..10000</a>
%F A344335 a(n) = A000005(A001016(n)).
%F A344335 Multiplicative with a(p^e) = 8*e+1.
%F A344335 a(n) = Sum_{d|n} 8^omega(d).
%F A344335 G.f.: Sum_{k>=1} 8^omega(k) * x^k/(1 - x^k).
%F A344335 Dirichlet g.f.: zeta(s)^2 * Product_{primes p} (1 + 7/p^s). - _Vaclav Kotesovec_, Aug 19 2021
%t A344335 Table[DivisorSigma[0, n^8], {n, 1, 100}] (* _Amiram Eldar_, May 15 2021 *)
%o A344335 (PARI) a(n) = numdiv(n^8);
%o A344335 (PARI) a(n) = prod(k=1, #f=factor(n)[, 2], 8*f[k]+1);
%o A344335 (PARI) a(n) = sumdiv(n, d, 8^omega(d));
%o A344335 (PARI) my(N=99, x='x+O('x^N)); Vec(sum(k=1, N, 8^omega(k)*x^k/(1-x^k)))
%o A344335 (PARI) for(n=1, 100, print1(direuler(p=2, n, (1 + 7*X)/(1 - X)^2)[n], ", ")) \\ _Vaclav Kotesovec_, Aug 19 2021
%Y A344335 Column k=8 of A343656.
%Y A344335 Cf. A000005, A001016.
%K A344335 nonn,mult
%O A344335 1,2
%A A344335 _Seiichi Manyama_, May 15 2021