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A036541 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!.

%I A036541 #26 May 19 2018 02:12:32
%S A036541 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 A036541 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 A036541 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 A036541 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 A036541 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 A036541 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 A036541 Robert Israel, <a href="/A036541/b036541.txt">Table of n, a(n) for n = 1..10000</a>
%F A036541 a(n) = omega(n!) - omega(binomial(n, floor(n/2))) = PrimePi(n) - omega(binomial(n, floor(n/2))).
%e A036541 a(1000) = PrimePi(1000) - omega(binomial(1000, 500)) = 168 - 116 = 52.
%p A036541 N:= 1000: # to get a(1) .. a(N) G:= proc(p,n) local m,Ln,Lm;
%p A036541    m:= floor(n/2);
%p A036541    Ln:= convert(n,base,p);
%p A036541    Lm:= convert(m,base,p);
%p A036541    hastype(Ln[1..nops(Lm)]-Lm,negative)
%p A036541 end proc:
%p A036541 S[1]:= {}:
%p A036541 S[2]:= {}:
%p A036541 for n from 3 to N do
%p A036541   if n::even then
%p A036541      if n = 2^ilog2(n) then S[n]:= S[n-1] minus {2}
%p A036541      else S[n]:= S[n-1]
%p A036541      fi
%p A036541   else
%p A036541      S[n]:= (S[n-1] minus select(G,numtheory:-factorset(n),n)) union remove(G,numtheory:-factorset((n+1)/2),n);
%p A036541   fi;
%p A036541 od:
%p A036541 seq(nops(S[i]),i=1..N); # _Robert Israel_, May 31 2016
%t A036541 Table[PrimePi@ n - PrimeNu[Binomial[n, Floor[n/2]]], {n, 105}] (* _Michael De Vlieger_, Jun 01 2016 *)
%Y A036541 Cf. A001405, A000720, A034973, A034974.
%K A036541 nonn
%O A036541 1,7
%A A036541 _Labos Elemer_