A356574 a(n) = Sum_{d|n} tau(d^4), where tau(n) = number of divisors of n, cf. A000005.

1, 6, 6, 15, 6, 36, 6, 28, 15, 36, 6, 90, 6, 36, 36, 45, 6, 90, 6, 90, 36, 36, 6, 168, 15, 36, 28, 90, 6, 216, 6, 66, 36, 36, 36, 225, 6, 36, 36, 168, 6, 216, 6, 90, 90, 36, 6, 270, 15, 90, 36, 90, 6, 168, 36, 168, 36, 36, 6, 540, 6, 36, 90, 91, 36, 216, 6, 90, 36, 216, 6, 420, 6, 36, 90, 90

Offset: 1

Author

Seiichi Manyama, Dec 13 2022

Keywords

• nonn
• easy
• mult

Links

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

Crossrefs

Cf. A007425, A035116, A061391, A358380, A359037, A359038.

Cf. A000005, A321348.

Programs

Mathematica

Array[DivisorSum[#, DivisorSigma[0, #^4] &] &, 120] (* Michael De Vlieger, Dec 13 2022 *)
f[p_, e_] := 2*e^2 + 3*e + 1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Dec 14 2022 *)

PARI

a(n) = sumdiv(n, d, numdiv(d^4));

PARI

a(n) = sumdiv(n, d, numdiv(n*d^2));

PARI

a(n) = sumdiv(n, d, numdiv(n^2));

PARI

a(n) = numdiv(n)*numdiv(n^2);

PARI

my(N=80, x='x+O('x^N)); Vec(sum(k=1, N, numdiv(k^4)*x^k/(1-x^k)))

Python

from math import prod
from sympy import factorint
def A356574(n): return prod((e+1)*((e<<1)+1) for e in factorint(n).values()) # Chai Wah Wu, Dec 13 2022

Formula

a(n) = Sum_{d|n} tau(n * d^2) = Sum_{d|n} tau(n^2).

a(n) = tau(n) * tau(n^2).

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

Multiplicative with a(p^e) = 2*e^2 + 3*e + 1. - Amiram Eldar, Dec 14 2022