A344327 Number of divisors of n^4.

1, 5, 5, 9, 5, 25, 5, 13, 9, 25, 5, 45, 5, 25, 25, 17, 5, 45, 5, 45, 25, 25, 5, 65, 9, 25, 13, 45, 5, 125, 5, 21, 25, 25, 25, 81, 5, 25, 25, 65, 5, 125, 5, 45, 45, 25, 5, 85, 9, 45, 25, 45, 5, 65, 25, 65, 25, 25, 5, 225, 5, 25, 45, 25, 25, 125, 5, 45, 25, 125, 5, 117, 5, 25, 45, 45, 25, 125, 5, 85, 17, 25

Offset

1,2

Links

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

Formula

a(n) = A000005(A000583(n)).

Multiplicative with a(p^e) = 4*e+1.

a(n) = Sum_{d|n} 4^omega(d).

G.f.: Sum_{k>=1} 4^omega(k) * x^k/(1 - x^k).

Dirichlet g.f.: zeta(s)^2 * Product_{primes p} (1 + 3/p^s). - Vaclav Kotesovec, May 15 2021

Dirichlet g.f.: zeta(s)^5 * Product_{primes p} (1 - 6/p^(2*s) + 8/p^(3*s) - 3/p^(4*s)). - Vaclav Kotesovec, Aug 20 2021

Mathematica

Table[DivisorSigma[0, n^4], {n, 1, 100}] (* Amiram Eldar, May 15 2021 *)

Prog

(PARI) a(n) = numdiv(n^4);
(PARI) a(n) = prod(k=1, #f=factor(n)[, 2], 4*f[k]+1);
(PARI) a(n) = sumdiv(n, d, 4^omega(d));
(PARI) my(N=99, x='x+O('x^N)); Vec(sum(k=1, N, 4^omega(k)*x^k/(1-x^k)))
(PARI) for(n=1, 100, print1(direuler(p=2, n, (1 + 3*X)/(1 - X)^2)[n], ", ")) \\ Vaclav Kotesovec, May 15 2021
(PARI) for(n=1, 100, print1(direuler(p=2, n, (1 - 6*X^2 + 8*X^3 - 3*X^4)/(1 - X)^5)[n], ", ")) \\ Vaclav Kotesovec, Aug 20 2021

Crossrefs

Column k=4 of A343656.

Cf. A000005, A000583, A202994.

Sequence in context: A199155 A243447 A046600 * A030798 A331502 A021951

Adjacent sequences: A344324 A344325 A344326 * A344328 A344329 A344330

Keyword

nonn,mult

Author

Seiichi Manyama, May 15 2021

Status

approved