A175417 Exponent of 2 minus sum of all other exponents, in the prime power factorization of n!

0, 0, 1, 0, 2, 1, 1, 0, 3, 1, 1, 0, 1, 0, 0, -2, 2, 1, 0, -1, 0, -2, -2, -3, -1, -3, -3, -6, -5, -6, -7, -8, -3, -5, -5, -7, -7, -8, -8, -10, -8, -9, -10, -11, -10, -13, -13, -14, -11, -13, -14, -16, -15, -16, -18, -20, -18, -20, -20, -21, -21, -22, -22, -25, -19, -21, -22

Offset: 0

Author

Zak Seidov, May 08 2010

Keywords

• sign

Comments

a(n)=0 for n={0,1,3,7,11,13,14,18,20}.

Examples

			a(20) = 0 because 20! = 2432902008176640000 = ((2^18)*(3^8)*(5^4)*(7^2)*(11^1)*(13^1)*(17^1)*(19^1)) and 18-(8+4+2+1+1+1+1) = 0.
		

Links

• Robert Israel, Table of n, a(n) for n = 0..10000

Crossrefs

Cf. A000142, A011371, A027868, A054861, A054896.

Programs

Maple

f:= proc(n) local t,p,k;
      p:= 2: t:= add(floor(n/2^k),k=1..ilog2(n)):
      do
        p:= nextprime(p);
        if n < p then return t fi;
        t:= t - add(floor(n/p^k),k=1..ilog[p](n))
      od
end proc:
map(f, [$0..100]); # Robert Israel, Nov 10 2024

Mathematica

Table[2*IntegerExponent[m!,2]-Total[Last/@FactorInteger[m! ]],{m,0,130}]

Formula

a(n)=2*A011371(n)-A022559(n).