A359037 a(n) = Sum_{d|n} tau(d^6), where tau(n) = number of divisors of n, cf. A000005.

1, 8, 8, 21, 8, 64, 8, 40, 21, 64, 8, 168, 8, 64, 64, 65, 8, 168, 8, 168, 64, 64, 8, 320, 21, 64, 40, 168, 8, 512, 8, 96, 64, 64, 64, 441, 8, 64, 64, 320, 8, 512, 8, 168, 168, 64, 8, 520, 21, 168, 64, 168, 8, 320, 64, 320, 64, 64, 8, 1344, 8, 64, 168, 133, 64, 512, 8, 168, 64, 512, 8, 840, 8

Offset

1,2

Links

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

Formula

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

a(n) = tau(n) * tau(n^3).

G.f.: Sum_{k>=1} tau(k^6) * x^k/(1 - x^k).

Multiplicative with a(p^e) = 3*e^2 + 4*e + 1. - Amiram Eldar, Dec 14 2022

Mathematica

Array[DivisorSum[#, DivisorSigma[0, #^6] &] &, 120] (* Michael De Vlieger, Dec 13 2022 *)
f[p_, e_] := 3*e^2 + 4*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^6));
(PARI) a(n) = sumdiv(n, d, numdiv(n*d^4));
(PARI) a(n) = sumdiv(n, d, numdiv(n^2*d^2));
(PARI) a(n) = sumdiv(n, d, numdiv(n^3));
(PARI) a(n) = numdiv(n)*numdiv(n^3);
(PARI) my(N=80, x='x+O('x^N)); Vec(sum(k=1, N, numdiv(k^6)*x^k/(1-x^k)))

Crossrefs

Cf. A007425, A035116, A061391, A356574, A358380, A359038.

Cf. A000005, A321348.

Sequence in context: A299592 A339707 A061156 * A195862 A227107 A205382

Adjacent sequences: A359034 A359035 A359036 * A359038 A359039 A359040

Keyword

nonn,mult,easy

Author

Seiichi Manyama, Dec 13 2022

Status

approved