The On-Line Encyclopedia of Integer Sequences (OEIS)

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A081389

Number of non-unitary prime divisors of Catalan numbers, i.e., number of those prime factors whose exponent is greater than one.
(history; published version)

#13 by N. J. A. Sloane at Sat Jul 29 19:24:58 EDT 2017

STATUS

proposed

approved

#12 by Jon E. Schoenfield at Sat Jul 29 16:50:12 EDT 2017

STATUS

editing

proposed

#11 by Jon E. Schoenfield at Sat Jul 29 16:50:07 EDT 2017

NAME

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

EXAMPLE

n=25: Catalan[25]=C[50,25]/26=4861946401452 =(2.2.3.3.7.7).29.31.37.41.43.47 unitary p-divisors = {29,31,37,41,43,47}, non-unitary ones = {2,3,7}, so a(25)=3.

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.

CROSSREFS

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

STATUS

approved

editing

#10 by Michel Marcus at Sun Feb 26 02:34:10 EST 2017

STATUS

reviewed

approved

#9 by Joerg Arndt at Sun Feb 26 02:27:52 EST 2017

STATUS

proposed

reviewed

#8 by Michel Marcus at Sun Feb 26 01:09:39 EST 2017

STATUS

editing

proposed

#7 by Michel Marcus at Sun Feb 26 01:08:46 EST 2017

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

#6 by Michel Marcus at Sun Feb 26 01:00:31 EST 2017

FORMULA

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

EXAMPLE

n=25: Catalan[25]=C[50,25]/26=4861946401452 =(2.2.3.3.7.7).29.31.37.41.43.47 unitary p-divisors = {29,31,37,41,43,47}, non-unitary ones = {2,3,7}, so a(25)=3/.

CROSSREFS

Cf. A000108, A080405, A081386-A081389, , A081387, A081388, A034444, A048105, A056170, A056169.

STATUS

proposed

editing

#5 by Michael De Vlieger at Sat Feb 25 22:48:09 EST 2017

STATUS

editing

proposed

#4 by Michael De Vlieger at Sat Feb 25 22:47:40 EST 2017

LINKS

Michael De Vlieger, <a href="/A081389/b081389.txt">Table of n, a(n) for n = 1..10000</a>

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 *)

STATUS

approved

editing