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

2,2

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

Links

Table of n, a(n) for n=2..60.

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

Cf. A001142, A109873.

Sequence in context: A034974 A048275 A211508 * A069345 A341948 A347661

Adjacent sequences: A109871 A109872 A109873 * A109875 A109876 A109877

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