A361265 Multiplicative with a(p^e) = e * p^(e + 1), e > 0.

1, 4, 9, 16, 25, 36, 49, 48, 54, 100, 121, 144, 169, 196, 225, 128, 289, 216, 361, 400, 441, 484, 529, 432, 250, 676, 243, 784, 841, 900, 961, 320, 1089, 1156, 1225, 864, 1369, 1444, 1521, 1200, 1681, 1764, 1849, 1936, 1350, 2116, 2209, 1152, 686, 1000, 2601, 2704

Offset

1,2

Links

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

Formula

Dirichlet g.f.: Product_{primes p} (1 + p^(s + 2)/(p^s - p)^2).

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

Sum_{k=1..n} a(k) ~ c * Pi^4 * n^3 / 108, where c = Product_{primes p} (1 - 3/p^2 + 2/p^3 + 1/p^4 - 1/p^5) = 0.3086489554825164955853322259998244718829914385...

a(n) = A005361(n) * A064549(n).

Sum_{n>=1} 1/a(n) = Product_{p prime} (1 - log(1-1/p))/p = 1.6843597117... . - Amiram Eldar, Sep 01 2023

Mathematica

g[p_, e_] := e*p^(e+1); 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^2 * X / (1 - p*X)^2)[n], ", "))

Crossrefs

Cf. A005361, A064549, A203639, A361268.

Sequence in context: A159852 A343066 A028907 * A292677 A292678 A072595

Adjacent sequences: A361262 A361263 A361264 * A361266 A361267 A361268

Keyword

nonn,easy,mult

Author

Vaclav Kotesovec, Mar 06 2023

Status

approved