A152765 Smallest prime divisor of Catalan number A000108(n), with a(0) = a(1) = 1.

1, 1, 2, 5, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 7, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2

Offset: 0

Author

Omar E. Pol, Dec 15 2008, Jan 03 2009

Keywords

• nonn

Comments

a(n) <> 2 iff n = 2^k - 1 (A000225). In fact for k>1, a(2^k-1): 5, 3, 3, 7, 3, 3, 7, 3, 3, 3, 3, 3, 3, ..., . (A120275) - Robert G. Wilson v, Nov 14 2015

Links

• Antti Karttunen, Table of n, a(n) for n = 0..65537

Crossrefs

Cf. A000108, A000225, A120275, A120303, A139044, A152761, A152762, A152763.

Programs

Magma

[Minimum(PrimeDivisors(Catalan(n))): n in [2..100]]; // Vincenzo Librandi, Jan 04 2017

Mathematica

FactorInteger[#][[1,1]]&/@CatalanNumber[Range[2,80]] (* Harvey P. Dale, Oct 08 2014 *)

PARI

a(n) = if (n<=1, 1, factor(binomial(2*n, n)/(n+1))[1, 1]); \\ Michel Marcus, Nov 14 2015; corrected Jun 13 2022

PARI

A152765(n) = if(n<2,1,my(c=binomial(2*n, n)/(n+1)); forprime(p=2, oo, if(!(c%p),return(p)))); \\ Antti Karttunen, Jan 12 2019

Formula

a(n) = A020639(A000108(n)). - Michel Marcus, Nov 14 2015

Extensions

Terms a(0) = a(1) = 1 prepended and more terms added by Antti Karttunen, Jan 12 2019