A359038 a(n) = Sum_{d|n} tau(d^7), where tau(n) = number of divisors of n, cf. A000005.

1, 9, 9, 24, 9, 81, 9, 46, 24, 81, 9, 216, 9, 81, 81, 75, 9, 216, 9, 216, 81, 81, 9, 414, 24, 81, 46, 216, 9, 729, 9, 111, 81, 81, 81, 576, 9, 81, 81, 414, 9, 729, 9, 216, 216, 81, 9, 675, 24, 216, 81, 216, 9, 414, 81, 414, 81, 81, 9, 1944, 9, 81, 216, 154, 81, 729, 9, 216, 81, 729, 9

Offset

1,2

Links

Table of n, a(n) for n=1..71.

Formula

a(n) = Sum_{d|n} tau(n * d^5) = Sum_{d|n} tau(n^2 * d^3) = Sum_{d|n} tau(n^3 * d) = Sum_{d|n} tau(n^4 / d).

G.f.: Sum_{k>=1} tau(k^7) * x^k/(1 - x^k).

Multiplicative with a(p^e) = 7*e^2/2 + 9*e/2 + 1. - Amiram Eldar, Dec 14 2022

Mathematica

Array[DivisorSum[#, DivisorSigma[0, #^7] &] &, 120] (* Michael De Vlieger, Dec 13 2022 *)
f[p_, e_] := 7*e^2/2 + 9*e/2 + 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^7));
(PARI) a(n) = sumdiv(n, d, numdiv(n*d^5));
(PARI) a(n) = sumdiv(n, d, numdiv(n^2*d^3));
(PARI) a(n) = sumdiv(n, d, numdiv(n^3*d));
(PARI) a(n) = sumdiv(n, d, numdiv(n^4/d));
(PARI) my(N=80, x='x+O('x^N)); Vec(sum(k=1, N, numdiv(k^7)*x^k/(1-x^k)))
(Python)
from math import prod
from sympy import factorint
def A359038(n): return prod((e+1)*(7*e+2)>>1 for e in factorint(n).values()) # Chai Wah Wu, Dec 13 2022

Crossrefs

Cf. A000005, A007425, A035116, A061391, A356574, A358380, A359037.

Cf. A321348.

Sequence in context: A197497 A053456 A144424 * A282269 A205380 A246035

Adjacent sequences: A359035 A359036 A359037 * A359039 A359040 A359041

Keyword

nonn,mult,easy

Author

Seiichi Manyama, Dec 13 2022

Status

approved