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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.

%I A081389 #13 Jul 29 2017 19:24:58
%S A081389 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,
%T A081389 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,
%U A081389 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
%N A081389 Number of non-unitary prime divisors of Catalan numbers, i.e., number of those prime factors whose exponent is greater than one.
%H A081389 Michael De Vlieger, <a href="/A081389/b081389.txt">Table of n, a(n) for n = 1..10000</a>
%F A081389 a(n) = A056170(A000108(n)).
%e A081389 For n=25: Catalan(25) = binomial(50,25)/26 = 4861946401452 =(2*2*3*3*7*7)*29*31*37*41*43*47;
%e A081389 unitary prime divisors: {29,31,37,41,43,47};
%e A081389 non-unitary prime divisors: {2,3,7}, so a(25) = 3.
%t A081389 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 *)
%o A081389 (PARI) catalan(n) = binomial(2*n, n)/(n+1);
%o A081389 nbud(n) = #select(x->x!=1, factor(n)[,2]);
%o A081389 a(n) = nbud(catalan(n)); \\ _Michel Marcus_, Feb 26 2017
%Y A081389 Cf. A000108, A034444, A048105, A056169, A056170, A080405, A081386, A081387, A081388.
%K A081389 nonn
%O A081389 1,13
%A A081389 _Labos Elemer_, Mar 27 2003