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A344329 Number of divisors of n^6.

%I A344329 #21 Aug 19 2021 12:55:40
%S A344329 1,7,7,13,7,49,7,19,13,49,7,91,7,49,49,25,7,91,7,91,49,49,7,133,13,49,
%T A344329 19,91,7,343,7,31,49,49,49,169,7,49,49,133,7,343,7,91,91,49,7,175,13,
%U A344329 91,49,91,7,133,49,133,49,49,7,637,7,49,91,37,49,343,7,91,49,343,7,247,7
%N A344329 Number of divisors of n^6.
%H A344329 Seiichi Manyama, <a href="/A344329/b344329.txt">Table of n, a(n) for n = 1..10000</a>
%F A344329 a(n) = A000005(A001014(n)).
%F A344329 Multiplicative with a(p^e) = 6*e+1.
%F A344329 a(n) = Sum_{d|n} 6^omega(d).
%F A344329 G.f.: Sum_{k>=1} 6^omega(k) * x^k/(1 - x^k).
%F A344329 Dirichlet g.f.: zeta(s)^2 * Product_{primes p} (1 + 5/p^s). - _Vaclav Kotesovec_, Aug 19 2021
%t A344329 Table[DivisorSigma[0, n^6], {n, 1, 100}] (* _Amiram Eldar_, May 15 2021 *)
%o A344329 (PARI) a(n) = numdiv(n^6);
%o A344329 (PARI) a(n) = prod(k=1, #f=factor(n)[, 2], 6*f[k]+1);
%o A344329 (PARI) a(n) = sumdiv(n, d, 6^omega(d));
%o A344329 (PARI) my(N=99, x='x+O('x^N)); Vec(sum(k=1, N, 6^omega(k)*x^k/(1-x^k)))
%o A344329 (PARI) for(n=1, 100, print1(direuler(p=2, n, (1 + 5*X)/(1 - X)^2)[n], ", ")) \\ _Vaclav Kotesovec_, Aug 19 2021
%Y A344329 Column k=6 of A343656.
%Y A344329 Cf. A000005, A001014.
%K A344329 nonn,mult
%O A344329 1,2
%A A344329 _Seiichi Manyama_, May 15 2021