This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A147844 #24 Feb 02 2025 09:36:01
%S A147844 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,
%T A147844 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,
%U A147844 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
%N 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.
%C A147844 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.
%H A147844 MathForFun, <a href="http://groups.yahoo.com/group/mathforfun/message/13576">Binomial Identity</a>.
%H A147844 Kenneth Ramsey, Cino Hilliard and others, <a href="/A147844/a147844.txt">Binomial Identity ..Hard to Prove this?</a>, digest of 6 messages in mathforfun Yahoo group, Oct 30 - Nov 10, 2008. [Cached copy]
%e A147844 (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).
%t A147844 Table[PrimePi[2n]-PrimeNu[(2n)!/(n!)^2],{n,100}] (* _Harvey P. Dale_, Oct 30 2021 *)
%o A147844 (PARI) g2(n) = for(x=1,n,ct=omega((2*x)!/x!^2);print1(primepi(2*x)-ct","))
%o A147844 (Magma) [#PrimesUpTo(2*n) - #PrimeDivisors( Factorial(2*n) div Factorial(n)^2):n in [1..91]]; // _Marius A. Burtea_, Nov 16 2019
%Y A147844 Cf. A001221, A001791, A099802.
%K A147844 nonn
%O A147844 1,7
%A A147844 _Cino Hilliard_, Nov 15 2008