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%I #12 Feb 03 2020 03:16:15
%S 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,
%T 10,11,10,8,6,12,13,14,13,11,14,15,16,13,11,10,12,18,20,19,20,11,13,
%U 19,22,18,15,26,20,17,17,26,21,22,18,18,23,26,20,19,23,21,22,19,27,17,35
%N Powers of 3 in the odd Catalan numbers Catalan(2^n - 1).
%C 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)).
%F a(n) = A007949(A038003(n)). - _Michel Marcus_, Feb 02 2020
%e 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.
%t 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} ]
%Y Cf. A007949, A038003.
%Y Cf. A000108, A048896, A048881.
%K nonn
%O 1,4
%A _Wouter Meeussen_, Mar 28 2001
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