OFFSET
1,4
COMMENTS
The logarithm of the Dirichlet series with the reciprocals of this sequence as coefficients is the Dirichlet series with the characteristic function of primes A010051 as coefficients. - Mats Granvik, Apr 13 2011
LINKS
Antti Karttunen, Table of n, a(n) for n = 1..10000
Rafael Jakimczuk, Generalizations of Mertens's Formula and k-Free and s-Full Numbers with Prime Divisors in Arithmetic Progression, ResearchGate, 2024.
FORMULA
From Antti Karttunen, May 29 2017: (Start)
From Mats Granvik, Mar 05 2019: (Start)
log(a(n)) = inverse Möbius transform of log(A306694(n)).
From Amiram Eldar, Mar 08 2024: (Start)
Let f(n) = 1/a(n). Formulas from Jakimczuk (2024, pp. 12-15):
Dirichlet g.f. of f(n): Sum_{n>=1} f(n)/n^s = exp(P(s)), where P(s) is the prime zeta function.
Sum_{k=1..n} f(k) = c * n + o(n), where c = A240953.
Sum_{k=1..n} f(k)/k = c * log(n) + o(log(n)), where c = A240953. (End)
EXAMPLE
45 = 3^2 * 5^1. So a(45) = 2! * 1! = 2.
MAPLE
w := n -> op(2, ifactors(n)): a := n -> mul(factorial(w(n)[j][2]), j = 1..nops(w(n))): seq(a(n), n = 1..101); # Emeric Deutsch, May 17 2012
MATHEMATICA
f[n_] := Block[{fi = Last@Transpose@FactorInteger@n}, Times @@ (fi!)]; Array[f, 101] (* Robert G. Wilson v, Dec 27 2005 *)
PROG
(PARI) A112624(n) = { my(f = factor(n), m = 1); for (k=1, #f~, m *= f[k, 2]!; ); m; } \\ Antti Karttunen, May 28 2017
(Sage)
def A112624(n):
return mul(factorial(s[1]) for s in factor(n))
[A112624(i) for i in (1..101)] # Peter Luschny, Jun 15 2013
CROSSREFS
KEYWORD
nonn,easy,mult
AUTHOR
Leroy Quet, Dec 25 2005
EXTENSIONS
More terms from Robert G. Wilson v, Dec 27 2005
STATUS
approved