%I #26 May 19 2018 02:12:32
%S 0,0,1,0,1,1,2,1,1,1,1,1,2,2,2,1,2,2,3,3,2,2,3,3,3,3,3,3,3,3,4,3,3,3,
%T 2,2,3,3,2,2,3,3,4,4,5,5,5,5,5,5,5,5,6,6,4,4,3,3,5,5,6,6,6,5,4,4,5,5,
%U 5,5,6,6,7,7,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,6,6,7,7,7,7,8,8,9,9,8
%N Deficit of central binomial coefficients in terms of number of prime factors: a(n) shows how many fewer prime factors the n-th central binomial coefficient has than n!.
%C Primes not exceeding n/2 are missing from this kit of prime divisors. Note differences of consecutive deficits change sign like: 0,1,0,-2,0,-1,0,+2,0.
%C a(2n) = a(2n-1) unless n = 2^k for some k >= 1, in which case a(2n) = a(2n-1)-1. - _Robert Israel_, May 31 2016
%H Robert Israel, <a href="/A036541/b036541.txt">Table of n, a(n) for n = 1..10000</a>
%F a(n) = omega(n!) - omega(binomial(n, floor(n/2))) = PrimePi(n) - omega(binomial(n, floor(n/2))).
%e a(1000) = PrimePi(1000) - omega(binomial(1000, 500)) = 168 - 116 = 52.
%p N:= 1000: # to get a(1) .. a(N) G:= proc(p,n) local m,Ln,Lm;
%p m:= floor(n/2);
%p Ln:= convert(n,base,p);
%p Lm:= convert(m,base,p);
%p hastype(Ln[1..nops(Lm)]-Lm,negative)
%p end proc:
%p S[1]:= {}:
%p S[2]:= {}:
%p for n from 3 to N do
%p if n::even then
%p if n = 2^ilog2(n) then S[n]:= S[n-1] minus {2}
%p else S[n]:= S[n-1]
%p fi
%p else
%p S[n]:= (S[n-1] minus select(G,numtheory:-factorset(n),n)) union remove(G,numtheory:-factorset((n+1)/2),n);
%p fi;
%p od:
%p seq(nops(S[i]),i=1..N); # _Robert Israel_, May 31 2016
%t Table[PrimePi@ n - PrimeNu[Binomial[n, Floor[n/2]]], {n, 105}] (* _Michael De Vlieger_, Jun 01 2016 *)
%Y Cf. A001405, A000720, A034973, A034974.
%K nonn
%O 1,7
%A _Labos Elemer_
|