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

1,2

Links

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

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

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 *)

Prog

(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

Crossrefs

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

Cf. A000005, A321348.

Sequence in context: A339721 A341832 A072695 * A330568 A085596 A107620

Adjacent sequences: A356571 A356572 A356573 * A356575 A356576 A356577

Keyword

nonn,easy,mult

Author

Seiichi Manyama, Dec 13 2022

Status

approved