A321348 a(n) = Sum_{d|n} tau(d^n), where tau() is the number of divisors (A000005).

1, 4, 5, 15, 7, 64, 9, 52, 30, 144, 13, 546, 15, 256, 289, 165, 19, 1140, 21, 1386, 529, 576, 25, 3848, 78, 784, 166, 2610, 31, 32768, 33, 486, 1225, 1296, 1369, 12321, 39, 1600, 1681, 10248, 43, 85184, 45, 6210, 6486, 2304, 49, 24250, 150, 7956

Offset: 1

Author

Ilya Gutkovskiy, Nov 06 2018

Keywords

• nonn

Comments

a(n) is prime iff n is in A001359, which makes the sequence a supersequence of A006512. - Ivan N. Ianakiev, Nov 07 2018

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..5000

Crossrefs

Cf. A000005, A007425, A035116, A061391.

Cf. A001359, A006512, A052147.

Programs

Magma

[&+[NumberOfDivisors(d^n): d in Divisors(n)]: n in [1..50]]; // Vincenzo Librandi, Nov 08 2018

Maple

with(numtheory): seq(coeff(series(add(tau(k^n)*x^k/(1-x^k),k=1..n),x,n+1), x, n), n = 1 .. 50); # Muniru A Asiru, Nov 25 2018

Mathematica

Table[Sum[DivisorSigma[0, d^n], {d, Divisors[n]}], {n, 50}]
a[n_] := Times @@ ((#[[2]] + 1) (n #[[2]] + 2)/2 & /@ FactorInteger[n]); a[1] = 1; Table[a[n], {n, 50}]

PARI

a(n) = sumdiv(n, d, numdiv(d^n)); \\ Michel Marcus, Nov 06 2018

Python

from math import prod
from sympy import factorint
def A321348(n): return prod((e+1)*(n*e+2)>>1 for e in factorint(n).values()) # Chai Wah Wu, Dec 13 2022

Formula

a(n) = [x^n] Sum_{k>=1} tau(k^n)*x^k/(1 - x^k).

If n = Product (p_j^k_j) then a(n) = Product ((k_j + 1)*(n*k_j + 2)/2).

a(prime(n)) = prime(n) + 2 = A052147(n). - Michel Marcus, Nov 25 2018