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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!.
1
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
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