A344328 Number of divisors of n^5.

1, 6, 6, 11, 6, 36, 6, 16, 11, 36, 6, 66, 6, 36, 36, 21, 6, 66, 6, 66, 36, 36, 6, 96, 11, 36, 16, 66, 6, 216, 6, 26, 36, 36, 36, 121, 6, 36, 36, 96, 6, 216, 6, 66, 66, 36, 6, 126, 11, 66, 36, 66, 6, 96, 36, 96, 36, 36, 6, 396, 6, 36, 66, 31, 36, 216, 6, 66, 36, 216, 6, 176, 6, 36, 66, 66, 36

Offset: 1

Author

Seiichi Manyama, May 15 2021

Keywords

• nonn
• mult

Links

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

Crossrefs

Column k=5 of A343656.

Cf. A000005, A000584, A082476 (5^omega(n)), A203556.

Programs

Mathematica

Table[DivisorSigma[0, n^5], {n, 1, 100}] (* Amiram Eldar, May 15 2021 *)

PARI

a(n) = numdiv(n^5);

PARI

a(n) = prod(k=1, #f=factor(n)[, 2], 5*f[k]+1);

PARI

a(n) = sumdiv(n, d, 5^omega(d));

PARI

my(N=99, x='x+O('x^N)); Vec(sum(k=1, N, 5^omega(k)*x^k/(1-x^k)))

PARI

for(n=1, 100, print1(direuler(p=2, n, (1 + 4*X)/(1 - X)^2)[n], ", ")) \\ Vaclav Kotesovec, Aug 19 2021

Formula

a(n) = A000005(A000584(n)).

Multiplicative with a(p^e) = 5*e+1.

a(n) = Sum_{d|n} 5^omega(d).

G.f.: Sum_{k>=1} 5^omega(k) * x^k/(1 - x^k).

Dirichlet g.f.: zeta(s)^2 * Product_{primes p} (1 + 4/p^s). - Vaclav Kotesovec, Aug 19 2021