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 A119861 Number of distinct prime factors of the odd Catalan numbers A038003(n). 4
 0, 1, 3, 6, 11, 20, 36, 64, 117, 209, 381, 699, 1291, 2387, 4445, 8317, 15645, 29494, 55855, 106021, 201778, 384941, 735909, 1409683, 2705277, 5200202 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 COMMENTS A038003[n] = A000108[2^n-1] = binomial(2^(n+1)-2, 2^n-1)/(2^n). a(1) = 0 because A038003[1] = 1. a(2) = 1 because A038003[2] = 5. a(3) = 3 because A038003[3] = 429 = 3*11*13. a(4) = 6 because A038003[4] = 9694845 = 3^2*5*17*19*23*29. Odd Catalan numbers are listed in A038003[n] = A000108[2^n-1] = binomial(2^(n+1)-2, 2^n-1)/(2^n). LINKS Eric Weisstein's World of Mathematics, Catalan Number. FORMULA a(n) = Length[ FactorInteger[ Binomial[ 2^(n+1)-2, 2^n-1] / (2^n) ]]. MAPLE with(numtheory): c:=proc(n) options operator, arrow: binomial(2*n, n)/(n+1) end proc: seq(nops(factorset(c(2^n-1))), n=1..15); # Emeric Deutsch, Oct 24 2007 MATHEMATICA Table[Length[FactorInteger[Binomial[2^(n+1)-2, 2^n-1]/(2^n)]], {n, 1, 15}] PROG (Python) from sympy import factorint A119861_list, c, s = [0], {}, 3 for n in range(2, 2**19): ....for p, e in factorint(4*n-2).items(): ........if p in c: ............c[p] += e ........else: ............c[p] = e ....for p, e in factorint(n+1).items(): ........if c[p] == e: ............del c[p] ........else: ............c[p] -= e ....if n == s: ........A119861_list.append(len(c)) ........s = 2*s+1 # Chai Wah Wu, Feb 12 2015 CROSSREFS Cf. A038003, A000108, A120274, A120275. Cf. A000108 = Catalan Number. Cf. A038003 = Odd Catalan numbers. Cf. A120274, A120275, A119908, A094389. Sequence in context: A077855 A054887 A019302 * A255061 A018075 A125896 Adjacent sequences:  A119858 A119859 A119860 * A119862 A119863 A119864 KEYWORD nonn AUTHOR Alexander Adamchuk, Jul 31 2006, Oct 11 2007 EXTENSIONS a(16)-a(18) from Robert G. Wilson v, May 15 2007 a(19)-a(26) from Chai Wah Wu, Feb 12 2015 STATUS approved

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Last modified December 11 10:48 EST 2018. Contains 318049 sequences. (Running on oeis4.)