A109874 Largest exponent e such that n^e that divides A001142(n).

1, 2, 2, 4, 4, 6, 5, 7, 8, 10, 9, 12, 11, 12, 12, 16, 14, 18, 16, 18, 20, 22, 19, 22, 24, 22, 23, 28, 26, 30, 25, 30, 32, 30, 36, 36, 36, 36, 40, 40, 36, 42, 40, 39, 44, 46, 40, 45, 44, 46, 48, 52, 45, 50, 49, 54, 56, 58, 54

Offset: 2

Author

Amarnath Murthy, Jul 10 2005

Keywords

• nonn

Comments

a(n) = n-1, if n is a prime. If n is composite, a(n) >= 2.

Conjectures: (1) If n is even and n = 2^r*m, m odd and >1, then a(n)= n-r-1. (2) If n = 2^r then a(n) = n-3. (3) If n is odd and composite then a(n) = n-2.

a(n) is the highest exponent e such that n^e divides Product_{k=0..n} binomial(n, k). - Joerg Arndt, Jun 04 2022

Crossrefs

Cf. A001142, A109873.

Programs

Maple

A001142 := proc(n) local k ; mul(k^(2*k-1-n),k=1..n) ; end: A109874 := proc(n) local a,k; a := A001142(n) ; k := 0 ; while a mod n = 0 and a > 1 do a := a/n ; k := k+1 ; od; RETURN(k) ; end: seq(A109874(n),n=2..60) ; # R. J. Mathar, Aug 15 2007

Mathematica

a[n_] := IntegerExponent[Product[Binomial[n, k], {k, 0, n}], n];
Table[a[n], {n, 2, 60}] (* Jean-François Alcover, Apr 02 2024 *)

PARI

for(n=2,60,print1(valuation(prod(k=0,n,binomial(n,k)),n),", ")); \\ Joerg Arndt, Jun 04 2022

Extensions

Corrected and extended by R. J. Mathar, Aug 15 2007

Name corrected by Joerg Arndt, Jun 04 2022