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A023986 Sum of exponents of primes in C(4n,2n) - sum of exponents of primes in C(2n,n).

%I A023986 #17 Jun 11 2025 09:13:59
%S A023986 0,1,1,2,3,1,2,5,3,5,6,4,6,6,3,4,6,5,4,6,5,8,10,7,9,9,7,9,8,7,10,12,
%T A023986 10,9,13,11,13,16,12,13,14,9,11,12,12,13,12,12,13,15,11,13,16,13,15,
%U A023986 17,16,19,19,16,15,17,18,15,18,17,19,19,13,17,19,17,18,18,15,19,21,18,17,20,19,19,22,19,22
%N A023986 Sum of exponents of primes in C(4n,2n) - sum of exponents of primes in C(2n,n).
%H A023986 Charles R Greathouse IV, <a href="/A023986/b023986.txt">Table of n, a(n) for n = 0..10000</a>
%F A023986 From _Amiram Eldar_, Jun 11 2025: (Start)
%F A023986 a(n) = A023834(n) - A023816(n).
%F A023986 a(n) = A022559(4*n) - 3*A022559(2*n) + 2*A022559(n). (End)
%t A023986 a[n_] := PrimeOmega[Binomial[4*n, 2*n]] - PrimeOmega[Binomial[2*n, n]]; Array[a, 100, 0] (* _Amiram Eldar_, Jun 11 2025 *)
%o A023986 (PARI) a(n) = my(v = binomial(4*n, 2*n)/binomial(2*n, n)); bigomega(numerator(v)) - bigomega(denominator(v)); \\ _Michel Marcus_, Sep 30 2013
%o A023986 (PARI) vp(n,p)=my(s);while(n\=p,s+=n);s
%o A023986 a(n)=my(s);forprime(p=2,4*n,s+=vp(4*n,p)-3*vp(2*n,p)+2*vp(n,p)); s \\ _Charles R Greathouse IV_, Sep 30 2013
%Y A023986 Cf. A001222, A022559, A023816, A023834.
%Y A023986 Cf. A023978, A023979, A023980, A023981, A023982, A023983, A023984, A023985, A023987, A023990, A023991, A023992, A023993.
%K A023986 nonn
%O A023986 0,4
%A A023986 _Clark Kimberling_
%E A023986 Name clarified, offset changed to 0 and a(0) prepended by _Amiram Eldar_, Jun 11 2025