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

0,4

Links

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

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)

Mathematica

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

Prog

(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

Crossrefs

Cf. A001222, A022559, A023816, A023834.

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

Sequence in context: A190257 A050981 A053452 * A306732 A109202 A362599

Adjacent sequences: A023983 A023984 A023985 * A023987 A023988 A023989

Keyword

nonn

Author

Clark Kimberling

Extensions

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

Status

approved