A081387 - OEIS

%I #17 Oct 05 2024 09:10:18

%S 0,0,1,0,2,1,1,1,1,1,1,1,2,3,2,1,2,2,2,2,1,1,3,3,3,3,2,1,1,1,1,2,2,2,

%T 2,1,1,2,3,2,3,2,3,3,2,3,3,4,3,2,2,2,2,1,1,2,1,1,2,2,3,3,3,2,4,2,2,3,

%U 3,3,3,4,4,5,4,4,2,3,3,2,2,2,4,3,3,4,3,4,5,4,2,2,2,3,5,5,5,5,3,2,3,2,3,3,3

%N Number of non-unitary prime divisors of central binomial coefficient, C(2n,n) = A000984(n), i.e., number of prime factors in C(2n,n) whose exponent is greater than one.

%H T. D. Noe, <a href="/A081387/b081387.txt">Table of n, a(n) for n = 1..10000</a>

%F a(n) = A056170(A000984(n)) = A001221(A000984(n)) - A081386(n) = A067434(n) - A081386(n).

%e For n = 14: binomial(28,14) = 40116600 = 2*2*2*3*3*3*5*5*17*19*23; unitary prime divisors: {17,19,23}; non-unitary prime divisors: {2,3,5}, so a(14) = 3.

%t Table[Count[Transpose[FactorInteger[Binomial[2n,n]]][[2]],_?(#>1&)],{n,110}] (* _Harvey P. Dale_, Oct 08 2012 *)

%o (PARI) a(n) = {my(e = factor(binomial(2*n, n))[, 2]); sum(k = 1, #e, e[k] > 1);} \\ _Amiram Eldar_, Oct 05 2024

%Y Cf. A000984, A034444, A048105, A056169, A056170, A067434, A081387, A081388, A081389.

%K nonn

%O 1,5

%A _Labos Elemer_, Mar 27 2003