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!.

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, 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, 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

Offset

1,7

Comments

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.

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

Links

Robert Israel, Table of n, a(n) for n = 1..10000

Formula

a(n) = omega(n!) - omega(binomial(n, floor(n/2))) = PrimePi(n) - omega(binomial(n, floor(n/2))).

Example

a(1000) = PrimePi(1000) - omega(binomial(1000, 500)) = 168 - 116 = 52.

Maple

N:= 1000: # to get a(1) .. a(N) G:= proc(p, n) local m, Ln, Lm;
   m:= floor(n/2);
   Ln:= convert(n, base, p);
   Lm:= convert(m, base, p);
   hastype(Ln[1..nops(Lm)]-Lm, negative)
end proc:
S[1]:= {}:
S[2]:= {}:
for n from 3 to N do
  if n::even then
     if n = 2^ilog2(n) then S[n]:= S[n-1] minus {2}
     else S[n]:= S[n-1]
     fi
  else
     S[n]:= (S[n-1] minus select(G, numtheory:-factorset(n), n)) union remove(G, numtheory:-factorset((n+1)/2), n);
  fi;
od:
seq(nops(S[i]), i=1..N); # Robert Israel, May 31 2016

Mathematica

Table[PrimePi@ n - PrimeNu[Binomial[n, Floor[n/2]]], {n, 105}] (* Michael De Vlieger, Jun 01 2016 *)

Crossrefs

Cf. A001405, A000720, A034973, A034974.

Sequence in context: A080356 A184167 A352085 * A176505 A338521 A036225

Adjacent sequences: A036538 A036539 A036540 * A036542 A036543 A036544

Keyword

nonn

Author

Labos Elemer

Status

approved