A060828 Size of the Sylow 3-subgroup of the symmetric group S_n.

1, 1, 1, 3, 3, 3, 9, 9, 9, 81, 81, 81, 243, 243, 243, 729, 729, 729, 6561, 6561, 6561, 19683, 19683, 19683, 59049, 59049, 59049, 1594323, 1594323, 1594323, 4782969, 4782969, 4782969, 14348907, 14348907, 14348907, 129140163, 129140163, 129140163, 387420489

Offset: 0

Author

Ahmed Fares (ahmedfares(AT)my-deja.com), Apr 30 2001

Keywords

• nonn
• easy

Examples

			a(3) = 3 because in S_3 the Sylow 3-subgroup is the subgroup generated by the 3-cycles (123) and (132), its order is 3.
		

Links

• Harry J. Smith, Table of n, a(n) for n = 0..200
• Tyler Ball, Tom Edgar, and Daniel Juda, Dominance Orders, Generalized Binomial Coefficients, and Kummer's Theorem, Mathematics Magazine, Vol. 87, No. 2, April 2014, pp. 135-143.
• Index entries for sequences related to groups

Crossrefs

Cf. A054861, A060818.

Programs

Mathematica

(* By the formula: *) Table[3^IntegerExponent[n!, 3], {n, 0, 40}] (* Bruno Berselli, Aug 05 2013 *)

PARI

for (n=0, 200, s=0; d=3; while (n>=d, s+=n\d; d*=3); write("b060828.txt", n, " ", 3^s)) \\ Harry J. Smith, Jul 12 2009

Sage

def A060828(n):
    A004128 = lambda n: A004128(n//3) + n if n > 0 else 0
    return 3^A004128(n//3)
[A060828(i) for i in (0..39)]  # Peter Luschny, Nov 16 2012

Formula

a(n) = 3^A054861(n) = 3^(floor(n/3) + floor(n/9) + floor(n/27) + floor(n/81) + ...).

a(n) = Product_{i=1..n} A038500(i). - Tom Edgar, Apr 30 2014

a(n) = 3^(n/2 + O(log n)). - Charles R Greathouse IV, Aug 05 2015

Extensions

More terms from N. J. A. Sloane, Jul 03 2008