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A344336 Number of divisors of n^9.

%I A344336 #18 Aug 19 2021 12:55:13
%S A344336 1,10,10,19,10,100,10,28,19,100,10,190,10,100,100,37,10,190,10,190,
%T A344336 100,100,10,280,19,100,28,190,10,1000,10,46,100,100,100,361,10,100,
%U A344336 100,280,10,1000,10,190,190,100,10,370,19,190,100,190,10,280,100,280,100,100,10,1900,10,100
%N A344336 Number of divisors of n^9.
%H A344336 Seiichi Manyama, <a href="/A344336/b344336.txt">Table of n, a(n) for n = 1..10000</a>
%F A344336 a(n) = A000005(A001017(n)).
%F A344336 Multiplicative with a(p^e) = 9*e+1.
%F A344336 a(n) = Sum_{d|n} 9^omega(d).
%F A344336 G.f.: Sum_{k>=1} 9^omega(k) * x^k/(1 - x^k).
%F A344336 Dirichlet g.f.: zeta(s)^2 * Product_{primes p} (1 + 8/p^s). - _Vaclav Kotesovec_, Aug 19 2021
%t A344336 Table[DivisorSigma[0, n^9], {n, 1, 100}] (* _Amiram Eldar_, May 15 2021 *)
%o A344336 (PARI) a(n) = numdiv(n^9);
%o A344336 (PARI) a(n) = prod(k=1, #f=factor(n)[, 2], 9*f[k]+1);
%o A344336 (PARI) a(n) = sumdiv(n, d, 9^omega(d));
%o A344336 (PARI) my(N=99, x='x+O('x^N)); Vec(sum(k=1, N, 9^omega(k)*x^k/(1-x^k)))
%o A344336 (PARI) for(n=1, 100, print1(direuler(p=2, n, (1 + 8*X)/(1 - X)^2)[n], ", ")) \\ _Vaclav Kotesovec_, Aug 19 2021
%Y A344336 Column k=9 of A343656.
%Y A344336 Cf. A000005, A001017, A344337 (9^omega(n)).
%K A344336 nonn,mult
%O A344336 1,2
%A A344336 _Seiichi Manyama_, May 15 2021