A147844 Difference between the number of distinct prime divisors of (2*n)!/n!^2 and pi(2*n), where pi(x) is the prime counting function.

0, 0, 1, 1, 1, 1, 2, 1, 2, 3, 2, 3, 3, 3, 3, 3, 3, 2, 3, 2, 3, 4, 5, 5, 5, 5, 6, 4, 3, 5, 6, 5, 4, 5, 5, 6, 7, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 6, 7, 7, 8, 9, 8, 8, 10, 10, 11, 10, 10, 9, 9, 9, 9, 9, 9, 9, 8, 9, 10, 11, 11, 10, 10, 10, 10, 11, 10, 10, 11, 10, 10, 11, 11, 12, 12, 11, 12, 12, 12, 13, 13

Offset

1,7

Comments

The expression (2*n)!/n!^2 is taken from C(2*n+1,n+1) - C(2*n,n) = (2*n)!/(n!^2*(n/(n+1)) = sum(k=1,n,C(n,k)*C(n,k-1)). This was posed in the Yahoo Group MathForFun, see link.

Links

Table of n, a(n) for n=1..91.

MathForFun, Binomial Identity.

Kenneth Ramsey, Cino Hilliard and others, Binomial Identity ..Hard to Prove this?, digest of 6 messages in mathforfun Yahoo group, Oct 30 - Nov 10, 2008. [Cached copy]

Example

(2*10)!/10!^2 = 184756 = 2*2*11*13*17*19 which has 5 distinct divisors. Pi(2*10) = 8. 8-5=3 = a(10).

Mathematica

Table[PrimePi[2n]-PrimeNu[(2n)!/(n!)^2], {n, 100}] (* Harvey P. Dale, Oct 30 2021 *)

Prog

(PARI) g2(n) = for(x=1, n, ct=omega((2*x)!/x!^2); print1(primepi(2*x)-ct", "))
(Magma) [#PrimesUpTo(2*n) - #PrimeDivisors( Factorial(2*n) div Factorial(n)^2):n in [1..91]]; // Marius A. Burtea, Nov 16 2019

Crossrefs

Cf. A001221, A001791, A099802.

Sequence in context: A191408 A115756 A067731 * A291985 A317192 A053735

Adjacent sequences: A147841 A147842 A147843 * A147845 A147846 A147847

Keyword

nonn

Author

Cino Hilliard, Nov 15 2008

Extensions

Corrected the link. - Cino Hilliard, Nov 18 2008

Status

approved