A163176 The n-th Minkowski number divided by the n-th factorial: a(n) = A053657(n)/n!.

1, 1, 4, 2, 48, 16, 576, 144, 3840, 768, 9216, 1536, 3870720, 552960, 442368, 55296, 26542080, 2949120, 2229534720, 222953472, 70071091200, 6370099200, 76441190400, 6370099200, 16694755983360, 1284211998720, 570760888320

Offset: 1

Author

Jonathan Sondow, Jul 24 2009

Keywords

• easy
• nonn

Comments

a(n) is an integer by Legendre's formula for the exponent of the highest power of a prime dividing n!.

a(2n-1) = n*a(2n) because A053657(2n) = 2*A053657(2n-1).

See A053657 for additional comments, references, and links.

Examples

			a(4) = A053657(4)/4! = 48/24 = 2.
		

Links

• F. Bencherif, Sur une propriété des polynômes de Stirling, 26th Journées Arithmétiques, July 6-10, 2009, Université Jean Monnet, Saint-Etienne, France.
• Farid Bencherif, Tarek Garici, On a property of Stirling polynomials, Publications de l'Institut Mathématique (2017), Vol. 102, Issue 116, pp. 149-153.
• J.-L. Chabert, Integer-valued polynomials on prime numbers and logarithm power expansion, European J. Combinatorics 28 (2007) 754-761.

Crossrefs

Cf. A000142 (n!), A053657.

Programs

Julia

using Primes
function A163176(n)
    function L(n, p, r)
        s, q = 0, p - r
        while q <= n
            s += div(n, q)
            q *= p
        end
    s end
    n < 2 && return 1
    P = primes(n)
    prod(p^(L(n-1, p, 1) - L(n, p, 0)) for p in P)
end
[A163176(n) for n in 1:27] |> println  # Peter Luschny, Feb 07 2021

Maple

A163176 := proc(n) local L,p;
L := proc(n,p,r) local q,s; q := p-r; s := 0;
do if q > n then break fi; s := s+iquo(n,q);
q := q*p od; s end; mul(p^(L(n-1,p,1)-L(n,p,0)),
p = select(isprime,[$2..n])) end: # Peter Luschny, Jul 26 2009
ser := series((y/(exp(y)-1))^x, y, 29):
c := n -> denom(coeff(ser, y, n-1)):
seq(c(n)/n!, n=1..27); # Peter Luschny, May 13 2019

Mathematica

a[n_] := (1/n!)*Product[ p^Sum[ Floor[ (n-1)/((p-1)*p^k) ], {k, 0, n}], {p, Select[ Range[2, n], PrimeQ]}]; Table[ a[n], {n, 1, 27}] (* Jean-François Alcover, Dec 07 2011 *)

Formula

a(n) = A053657(n)/A000142(n).

a(n) = (1/n!)*Product_{p prime} p^(Sum_{k>=0} ((n-1)/((p-1)p^k))).

a(n) = (1/n!)*denominator([y^(n-1)](y/(exp(y)-1))^x). - Peter Luschny, May 13 2019