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

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, 10, 9, 13, 11, 13, 16, 12, 13, 14, 9, 11, 12, 12, 13, 12, 12, 13, 15, 11, 13, 16, 13, 15, 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

Offset: 0

Author

Clark Kimberling

Keywords

• nonn

Links

• Charles R Greathouse IV, Table of n, a(n) for n = 0..10000

Crossrefs

Cf. A001222, A022559, A023816, A023834.

Cf. A023978, A023979, A023980, A023981, A023982, A023983, A023984, A023985, A023987, A023990, A023991, A023992, A023993.

Programs

Mathematica

a[n_] := PrimeOmega[Binomial[4*n, 2*n]] - PrimeOmega[Binomial[2*n, n]]; Array[a, 100, 0] (* Amiram Eldar, Jun 11 2025 *)

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

PARI

vp(n,p)=my(s);while(n\=p,s+=n);s
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

Formula

From Amiram Eldar, Jun 11 2025: (Start)

a(n) = A023834(n) - A023816(n).

a(n) = A022559(4*n) - 3*A022559(2*n) + 2*A022559(n). (End)

Extensions

Name clarified, offset changed to 0 and a(0) prepended by Amiram Eldar, Jun 11 2025