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A081388 Number of unitary prime divisors of the n-th Catalan number.

%I A081388 #14 Aug 16 2024 17:39:30
%S A081388 0,1,1,2,3,2,3,4,4,3,5,4,3,4,5,6,8,6,9,8,8,8,7,7,6,7,9,10,10,10,11,11,
%T A081388 12,12,13,12,12,11,13,12,13,13,12,14,14,13,15,14,15,15,15,15,18,17,17,
%U A081388 17,17,18,17,18,17,18,18,19,21,20,22,19,19,19,21,19,19,20,20,23,23,22
%N A081388 Number of unitary prime divisors of the n-th Catalan number.
%H A081388 Amiram Eldar, <a href="/A081388/b081388.txt">Table of n, a(n) for n = 1..10000</a>
%F A081388 a(n) = A056169(A000108(n)).
%e A081388 For n = 10: Catalan(10) = 16796 = 2^2*13*17*19, the unitary prime divisors are {13, 17, 19}, so a(10) = 3.
%t A081388 a[n_] := Count[FactorInteger[CatalanNumber[n]][[;;, 2]], 1]; a[1] = 0; Array[a, 100] (* _Amiram Eldar_, Aug 16 2024 *)
%o A081388 (PARI) nbupd(n) = my(f=factor(n)[, 2]); sum(i=1, #f, f[i]==1);
%o A081388 a(n) = nbupd(binomial(2*n, n)/(n+1)); \\ _Michel Marcus_, Sep 05 2017
%Y A081388 Cf. A000108, A034444, A048105, A056169, A056170, A080405, A081386, A081387, A081389.
%K A081388 nonn
%O A081388 1,4
%A A081388 _Labos Elemer_, Mar 27 2003