A112624 If p^b(p,n) is the highest power of the prime p dividing n, then a(n) = Product_{p|n} b(p,n)!.

1, 1, 1, 2, 1, 1, 1, 6, 2, 1, 1, 2, 1, 1, 1, 24, 1, 2, 1, 2, 1, 1, 1, 6, 2, 1, 6, 2, 1, 1, 1, 120, 1, 1, 1, 4, 1, 1, 1, 6, 1, 1, 1, 2, 2, 1, 1, 24, 2, 2, 1, 2, 1, 6, 1, 6, 1, 1, 1, 2, 1, 1, 2, 720, 1, 1, 1, 2, 1, 1, 1, 12, 1, 1, 2, 2, 1, 1, 1, 24, 24, 1, 1, 2, 1, 1, 1, 6, 1, 2, 1, 2, 1, 1, 1, 120, 1, 2, 2, 4, 1

Offset: 1

Author

Leroy Quet, Dec 25 2005

Keywords

• nonn
• easy
• mult

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

Examples

			45 = 3^2 * 5^1. So a(45) = 2! * 1! = 2.
		

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.
• Index entries for sequences computed from exponents in factorization of n.

Crossrefs

For row > 1: a(n) = row products of A100995(A126988), when neglecting zero elements.

Cf. A000012, A000142, A010051, A028234, A067029, A112622, A112623, A156552, A240953, A246660, A306694.

Programs

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 *)

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

Scheme

(define (A112624 n) (if (= 1 n) n (* (A000142 (A067029 n)) (A112624 (A028234 n))))) ;; Antti Karttunen, May 29 2017

Formula

From Antti Karttunen, May 29 2017: (Start)

a(1) = 1 and for n > 1, a(n) = A000142(A067029(n)) * a(A028234(n)).

a(n) = A246660(A156552(n)). (End)

From Mats Granvik, Mar 05 2019: (Start)

log(a(n)) = inverse Möbius transform of log(A306694(n)).

log(a(n)) = Sum_{k=1..n} [k|n]*log(A306694(n/k))*A000012(k). (End)

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)

Extensions

More terms from Robert G. Wilson v, Dec 27 2005