A361268 Multiplicative with a(p^e) = e * p^(e + 2), e > 0.

1, 8, 27, 32, 125, 216, 343, 96, 162, 1000, 1331, 864, 2197, 2744, 3375, 256, 4913, 1296, 6859, 4000, 9261, 10648, 12167, 2592, 1250, 17576, 729, 10976, 24389, 27000, 29791, 640, 35937, 39304, 42875, 5184, 50653, 54872, 59319, 12000, 68921, 74088, 79507, 42592

Offset

1,2

Comments

In general, if the function is multiplicative with a(p^e) = e*p^(e+m) where m > 0, then Dirichlet g.f.: Product_{primes p} (1 + p^(s + m + 1)/(p^s - p)^2).

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

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

Limit_{m->oo} c(m) = 6/Pi^2 = A059956.

Links

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

Formula

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

Dirichlet g.f.: zeta(s-3) * zeta(s-1)^2 * Product_{primes p} (1 - p^(5 - 3*s) + p^(2 - 2*s) + 2*p^(4 - 2*s) - p^(6 - 2*s) - 2*p^(1 - s)).

Sum_{k=1..n} a(k) ~ c * zeta(3)^2 * n^4 / 4, where c = Product_{primes p} (1 - 1/p^2 - 2/p^3 + 2/p^4 + 1/p^6 - 1/p^7) = 0.47448576370894461600229128319633117903859559137234612880645471185501089953...

a(n) = A005361(n) * A361264(n).

Sum_{n>=1} 1/a(n) = Product_{p prime} (1 - log(1-1/p))/p^2 = 1.24331517732028787738... . - Amiram Eldar, Sep 01 2023

Mathematica

g[p_, e_] := 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)^2)[n], ", "))

Crossrefs

Cf. A005361, A059956, A203639, A361264, A361265.

Sequence in context: A304291 A056824 A367804 * A355038 A297868 A051760

Adjacent sequences: A361265 A361266 A361267 * A361269 A361270 A361271

Keyword

nonn,easy,mult

Author

Vaclav Kotesovec, Mar 06 2023

Status

approved