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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

Table of n, a(n) for n=1..26.

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 21:15 EST 2017. Contains 295919 sequences.