A359530 Multiplicative with a(p^e) = (p + 4)^e.

1, 6, 7, 36, 9, 42, 11, 216, 49, 54, 15, 252, 17, 66, 63, 1296, 21, 294, 23, 324, 77, 90, 27, 1512, 81, 102, 343, 396, 33, 378, 35, 7776, 105, 126, 99, 1764, 41, 138, 119, 1944, 45, 462, 47, 540, 441, 162, 51, 9072, 121, 486, 147, 612, 57, 2058, 135, 2376, 161

Offset: 1

Author

Vaclav Kotesovec, Feb 26 2023

Keywords

• nonn
• mult

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A166589 (multiplicative with a(p^e) = (p-3)^e), A166586 (p-2), A003958 (p-1), A000027 (p), A003959 (p+1), A166590 (p+2), A166591 (p+3).

Programs

Mathematica

g[p_, e_] := (p + 4)^e; a[1] = 1; a[n_] := Times @@ g @@@ FactorInteger[n]; Array[a, 100]

PARI

for(n=1, 100, print1(direuler(p=2, n, 1/(1-p*X-4*X))[n], ", "))

Python

from math import prod
from sympy import factorint
def A359530(n): return prod((p+4)**e for p, e in factorint(n).items()) # Chai Wah Wu, Feb 26 2023

Formula

Dirichlet g.f.: Product_{primes p} 1 / (1 - p^(1-s) - 4*p^(-s)).

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

Sum_{k=1..n} a(k) has an average value 2*c*zeta(r-1) * n^r / (3*log(6)), where r = 1 + log(3)/log(2) = 2.5849625007211561814537389439478165... and c = Product_{primes p, p>2} (1 + 4/(p^r - p - 4)) = 1.5747380964592139...