OFFSET
2,2
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
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 *)
PROG
(PARI) for(n=2, 60, print1(valuation(prod(k=0, n, binomial(n, k)), n), ", ")); \\ Joerg Arndt, Jun 04 2022
CROSSREFS
KEYWORD
nonn
AUTHOR
Amarnath Murthy, Jul 10 2005
EXTENSIONS
Corrected and extended by R. J. Mathar, Aug 15 2007
Name corrected by Joerg Arndt, Jun 04 2022
STATUS
approved