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

1,2

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

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

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}]

Prog

(PARI) a(n) = sumdiv(n, d, numdiv(d^n)); \\ Michel Marcus, Nov 06 2018
(Magma) [&+[NumberOfDivisors(d^n): d in Divisors(n)]: n in [1..50]]; // Vincenzo Librandi, Nov 08 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

Crossrefs

Cf. A000005, A007425, A035116, A061391.

Cf. A001359, A006512, A052147.

Sequence in context: A225204 A363817 A308095 * A257311 A369790 A330857

Adjacent sequences: A321345 A321346 A321347 * A321349 A321350 A321351

Keyword

nonn

Author

Ilya Gutkovskiy, Nov 06 2018

Status

approved