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
KEYWORD
nonn
AUTHOR
STATUS
approved