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Number of divisors of n^8.
2

%I #16 Aug 19 2021 12:55:24

%S 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 81,25,153,9,729,9,41,81,81,81,289,9,81,81,225,9,729,9,153,153,81,9,

%U 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 Number of divisors of n^8.

%H Seiichi Manyama, <a href="/A344335/b344335.txt">Table of n, a(n) for n = 1..10000</a>

%F a(n) = A000005(A001016(n)).

%F Multiplicative with a(p^e) = 8*e+1.

%F a(n) = Sum_{d|n} 8^omega(d).

%F G.f.: Sum_{k>=1} 8^omega(k) * x^k/(1 - x^k).

%F Dirichlet g.f.: zeta(s)^2 * Product_{primes p} (1 + 7/p^s). - _Vaclav Kotesovec_, Aug 19 2021

%t Table[DivisorSigma[0, n^8], {n, 1, 100}] (* _Amiram Eldar_, May 15 2021 *)

%o (PARI) a(n) = numdiv(n^8);

%o (PARI) a(n) = prod(k=1, #f=factor(n)[, 2], 8*f[k]+1);

%o (PARI) a(n) = sumdiv(n, d, 8^omega(d));

%o (PARI) my(N=99, x='x+O('x^N)); Vec(sum(k=1, N, 8^omega(k)*x^k/(1-x^k)))

%o (PARI) for(n=1, 100, print1(direuler(p=2, n, (1 + 7*X)/(1 - X)^2)[n], ", ")) \\ _Vaclav Kotesovec_, Aug 19 2021

%Y Column k=8 of A343656.

%Y Cf. A000005, A001016.

%K nonn,mult

%O 1,2

%A _Seiichi Manyama_, May 15 2021