A344335 Number of divisors of n^8.

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, 81, 25, 153, 9, 729, 9, 41, 81, 81, 81, 289, 9, 81, 81, 225, 9, 729, 9, 153, 153, 81, 9, 297, 17, 153, 81, 153, 9, 225, 81, 225, 81, 81, 9, 1377, 9, 81, 153, 49, 81, 729, 9, 153, 81, 729, 9

Offset

1,2

Links

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Formula

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

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

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

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

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

Mathematica

Table[DivisorSigma[0, n^8], {n, 1, 100}] (* Amiram Eldar, May 15 2021 *)

Prog

(PARI) a(n) = numdiv(n^8);
(PARI) a(n) = prod(k=1, #f=factor(n)[, 2], 8*f[k]+1);
(PARI) a(n) = sumdiv(n, d, 8^omega(d));
(PARI) my(N=99, x='x+O('x^N)); Vec(sum(k=1, N, 8^omega(k)*x^k/(1-x^k)))
(PARI) for(n=1, 100, print1(direuler(p=2, n, (1 + 7*X)/(1 - X)^2)[n], ", ")) \\ Vaclav Kotesovec, Aug 19 2021

Crossrefs

Column k=8 of A343656.

Cf. A000005, A001016.

Sequence in context: A350919 A144418 A003885 * A168390 A333151 A321659

Adjacent sequences: A344332 A344333 A344334 * A344336 A344337 A344338

Keyword

nonn,mult

Author

Seiichi Manyama, May 15 2021

Status

approved