A081389 Number of non-unitary prime divisors of Catalan numbers, i.e., number of those prime factors whose exponent is greater than one.

0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 2, 2, 1, 1, 0, 2, 0, 1, 1, 1, 2, 2, 3, 2, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 2, 1, 2, 2, 2, 3, 2, 2, 3, 2, 3, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 2, 2, 3, 2, 2, 2, 1, 2, 1, 3, 3, 3, 2, 4, 4, 4, 4, 2, 2, 3, 1, 1, 2, 2, 3, 2, 3, 3, 2, 4, 4, 2, 2, 2, 2, 3, 4, 5, 4, 3, 2, 2, 2, 2, 2, 2, 3

Offset: 1

Author

Labos Elemer, Mar 27 2003

Keywords

• nonn

Examples

			For n=25: Catalan(25) = binomial(50,25)/26 = 4861946401452 =(2*2*3*3*7*7)*29*31*37*41*43*47;
unitary prime divisors: {29,31,37,41,43,47};
non-unitary prime divisors: {2,3,7}, so a(25) = 3.
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000108, A034444, A048105, A056169, A056170, A080405, A081386, A081387, A081388.

Programs

Mathematica

Table[Boole[n == 1] + PrimeNu@ # - Count[Transpose[FactorInteger@ #][[2]], 1] &@ CatalanNumber@ n, {n, 105}] (* Michael De Vlieger, Feb 25 2017, after Harvey P. Dale at A056169 *)

PARI

catalan(n) = binomial(2*n, n)/(n+1);
nbud(n) = #select(x->x!=1, factor(n)[,2]);
a(n) = nbud(catalan(n)); \\ Michel Marcus, Feb 26 2017

Formula

a(n) = A056170(A000108(n)).