A119861 Number of distinct prime factors of the n-th odd Catalan number, A038003(n).

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

Offset

1,3

Links

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

Eric Weisstein's World of Mathematics, Catalan Number.

Formula

From Amiram Eldar, Jan 09 2026: (Start)

a(n) = A001221(A038003(n)).

a(n) = A080405(2^n-1). (End)

Example

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.

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}]
a[n_] := PrimeNu[CatalanNumber[2^n-1]]; Array[a, 15] (* Amiram Eldar, Jan 09 2026 *)

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. A000108 (Catalan numbers), A001221, A038003 (Odd Catalan numbers), A080405, A094389, A119908, A120274, A120275.

Sequence in context: A077855 A054887 A019302 * A255061 A018075 A125896

Adjacent sequences: A119858 A119859 A119860 * A119862 A119863 A119864

Keyword

nonn,more

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