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%I #40 Aug 19 2021 12:55:32
%S 1,8,8,15,8,64,8,22,15,64,8,120,8,64,64,29,8,120,8,120,64,64,8,176,15,
%T 64,22,120,8,512,8,36,64,64,64,225,8,64,64,176,8,512,8,120,120,64,8,
%U 232,15,120,64,120,8,176,64,176,64,64,8,960,8,64,120,43,64,512,8,120,64,512,8
%N Number of divisors of n^7.
%H Seiichi Manyama, <a href="/A343526/b343526.txt">Table of n, a(n) for n = 1..10000</a>
%F a(n) = A000005(A001015(n)).
%F Multiplicative with a(p^e) = 7*e+1.
%F a(n) = Sum_{d|n} 7^omega(d).
%F G.f.: Sum_{k>=1} 7^omega(k) * x^k/(1 - x^k).
%F Dirichlet g.f.: zeta(s)^2 * Product_{primes p} (1 + 6/p^s). - _Vaclav Kotesovec_, Aug 19 2021
%t Table[DivisorSigma[0, n^7], {n, 1, 100}] (* _Amiram Eldar_, May 15 2021 *)
%o (PARI) a(n) = numdiv(n^7);
%o (PARI) a(n) = prod(k=1, #f=factor(n)[, 2], 7*f[k]+1);
%o (PARI) a(n) = sumdiv(n, d, 7^omega(d));
%o (PARI) my(N=99, x='x+O('x^N)); Vec(sum(k=1, N, 7^omega(k)*x^k/(1-x^k)))
%o (PARI) for(n=1, 100, print1(direuler(p=2, n, (1 + 6*X)/(1 - X)^2)[n], ", ")) \\ _Vaclav Kotesovec_, Aug 19 2021
%Y Column k=7 of A343656.
%Y Cf. A000005, A001015.
%K nonn,mult
%O 1,2
%A _Seiichi Manyama_, May 15 2021
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