A344336 - OEIS

%I #18 Aug 19 2021 12:55:13

%S 1,10,10,19,10,100,10,28,19,100,10,190,10,100,100,37,10,190,10,190,

%T 100,100,10,280,19,100,28,190,10,1000,10,46,100,100,100,361,10,100,

%U 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 Number of divisors of n^9.

%H Seiichi Manyama, <a href="/A344336/b344336.txt">Table of n, a(n) for n = 1..10000</a>

%F a(n) = A000005(A001017(n)).

%F Multiplicative with a(p^e) = 9*e+1.

%F a(n) = Sum_{d|n} 9^omega(d).

%F G.f.: Sum_{k>=1} 9^omega(k) * x^k/(1 - x^k).

%F Dirichlet g.f.: zeta(s)^2 * Product_{primes p} (1 + 8/p^s). - _Vaclav Kotesovec_, Aug 19 2021

%t Table[DivisorSigma[0, n^9], {n, 1, 100}] (* _Amiram Eldar_, May 15 2021 *)

%o (PARI) a(n) = numdiv(n^9);

%o (PARI) a(n) = prod(k=1, #f=factor(n)[, 2], 9*f[k]+1);

%o (PARI) a(n) = sumdiv(n, d, 9^omega(d));

%o (PARI) my(N=99, x='x+O('x^N)); Vec(sum(k=1, N, 9^omega(k)*x^k/(1-x^k)))

%o (PARI) for(n=1, 100, print1(direuler(p=2, n, (1 + 8*X)/(1 - X)^2)[n], ", ")) \\ _Vaclav Kotesovec_, Aug 19 2021

%Y Column k=9 of A343656.

%Y Cf. A000005, A001017, A344337 (9^omega(n)).

%K nonn,mult

%O 1,2

%A _Seiichi Manyama_, May 15 2021