A081389 - OEIS

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

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OFFSET

1,13

LINKS

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

FORMULA

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

EXAMPLE

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.

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 *)

PROG

(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

CROSSREFS

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

Sequence in context: A123736 A358476 A185304 * A133685 A281492 A112183

Adjacent sequences: A081386 A081387 A081388 * A081390 A081391 A081392

KEYWORD

nonn

AUTHOR

Labos Elemer, Mar 27 2003

STATUS

approved