A060318 Powers of 3 in the odd Catalan numbers Catalan(2^n - 1).

0, 0, 1, 2, 0, 1, 3, 0, 3, 3, 3, 6, 2, 2, 9, 5, 5, 4, 8, 5, 9, 10, 5, 4, 4, 4, 9, 9, 8, 11, 13, 13, 10, 11, 10, 8, 6, 12, 13, 14, 13, 11, 14, 15, 16, 13, 11, 10, 12, 18, 20, 19, 20, 11, 13, 19, 22, 18, 15, 26, 20, 17, 17, 26, 21, 22, 18, 18, 23, 26, 20, 19, 23, 21, 22, 19, 27, 17, 35

Offset: 1

Author

Wouter Meeussen, Mar 28 2001

Keywords

• nonn

Comments

Conjecture: all odd Catalan numbers have smallest prime factor 3, except Catalan(3), which has smallest prime factor 5, and Catalan(31) and Catalan(255), which have smallest prime factor 7 (checked up to Catalan(-1 + 2^2048)).

Examples

			a(5)=0 because 2^5 -1 = 31 and Catalan(31) = 7*11*17*19*37*41*43*47*53*59*61 so the power of 3 is zero.
		

Crossrefs

Cf. A007949, A038003.

Cf. A000108, A048896, A048881.

Programs

Mathematica

pow3[ nfac_ ] := (nfac - Plus @@ IntegerDigits[ nfac, 3 ])/(3-1); powcat3[ n_ ] := pow3[ 2n ]-pow3[ n+1 ]-pow3[ n ]; Table[ powcat3[ 2^n-1 ], {n, 2048} ]

Formula

a(n) = A007949(A038003(n)). - Michel Marcus, Feb 02 2020