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A119861
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Number of distinct prime factors of the odd Catalan numbers A038003(n).
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4
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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
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OFFSET
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1,3
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COMMENTS
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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).
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LINKS
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FORMULA
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a(n) = Length[ FactorInteger[ Binomial[ 2^(n+1)-2, 2^n-1] / (2^n) ]].
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MAPLE
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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
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MATHEMATICA
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Table[Length[FactorInteger[Binomial[2^(n+1)-2, 2^n-1]/(2^n)]], {n, 1, 15}]
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PROG
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(Python)
from sympy import factorint
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))
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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