A361264 Multiplicative with a(p^e) = p^(e + 2), e > 0.

1, 8, 27, 16, 125, 216, 343, 32, 81, 1000, 1331, 432, 2197, 2744, 3375, 64, 4913, 648, 6859, 2000, 9261, 10648, 12167, 864, 625, 17576, 243, 5488, 24389, 27000, 29791, 128, 35937, 39304, 42875, 1296, 50653, 54872, 59319, 4000, 68921, 74088, 79507, 21296, 10125

Offset

1,2

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000

Formula

Dirichlet g.f.: Product_{primes p} (1 + p^3/(p^s - p)).

Dirichlet g.f.: zeta(s-3) * zeta(s-1) * Product_{primes p} (1 + p^(4-2*s) - p^(6-2*s) - p^(1-s)).

Sum_{k=1..n} a(k) ~ c * zeta(3) * n^4 / 4, where c = Product_{primes p} (1 - 1/p^2 - 1/p^3 + 1/p^4) = A330523 = 0.53589615382833799980850263131854595064822237...

From Amiram Eldar, Sep 01 2023: (Start)

a(n) = n * A007947(n)^2 = A064549(n) * A007947(n) = A064549(A064549(n)).

A000005(a(n)) = A360997(n).

Sum_{n>=1} 1/a(n) = Product_{p prime} (1 + 1/(p^2*(p-1))) = A065483. (End)

Mathematica

g[p_, e_] := p^(e+2); a[1] = 1; a[n_] := Times @@ g @@@ FactorInteger[n]; Array[a, 100]

Prog

(PARI) for(n=1, 100, print1(direuler(p=2, n, 1 + p^3*X/(1 - p*X))[n], ", "))

Crossrefs

Cf. A000005, A003557, A064549, A065483, A007947, A330523, A360997, A361266.

Sequence in context: A250140 A070510 A224787 * A070509 A070508 A070507

Adjacent sequences: A361261 A361262 A361263 * A361265 A361266 A361267

Keyword

nonn,easy,mult

Author

Vaclav Kotesovec, Mar 06 2023

Status

approved