A055656 - OEIS

%I #18 Jul 15 2024 10:23:56

%S 0,-1,0,0,2,2,3,3,3,3,4,4,6,6,6,6,10,10,11,11,11,11,12,12,12,12,12,12,

%T 14,14,15,15,15,15,15,15,17,17,17,17,20,20,21,21,21,21,22,22,22,22,22,

%U 22,24,24,24,24,24,24,25,25,27,27,27,27,27,27,28,28,28,28,29,29,32,32

%N Excess in exponents of powers of 2 in Euler phi of n! compared to that of n!.

%C The exponents of 2 is larger in phi(n!) than in n! if n > 4.

%H Amiram Eldar, <a href="/A055656/b055656.txt">Table of n, a(n) for n = 1..10000</a>

%F a(n) = A007814(A048855(n))-A007814(n!) = A007814(A048855(n))-A011371(A000142(n)).

%e For n = 8, 8! = 40320 = 128*315, phi(40320) = 9216 = 9*1024. The exponent of 2 in 8! is only 7, and in phi(8!) it is 10, so a(8) = 10-7 = 3.

%t eep2[n_]:=Module[{f=n!},IntegerExponent[EulerPhi[f],2]-IntegerExponent[f,2]]; Array[ eep2,80] (* _Harvey P. Dale_, Mar 18 2023 *)

%o (Python)

%o from math import factorial, prod

%o from sympy import primerange

%o from fractions import Fraction

%o def A055656(n): return (~(m:=((f:=factorial(n))*prod(Fraction(p-1,p) for p in primerange(n+1))).numerator)&m-1).bit_length()-(~f & f-1).bit_length() # _Chai Wah Wu_, Jul 06 2022

%o (PARI) a(n) = {my(f = n!); valuation(eulerphi(f), 2) - valuation(f, 2);} \\ _Amiram Eldar_, Jul 15 2024

%Y Cf. A000142, A000010, A048855, A007814, A011371.

%K sign

%O 1,5

%A _Labos Elemer_, Jul 11 2000