A055656 Excess in exponents of powers of 2 in Euler phi of n! compared to that of n!.

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, 14, 14, 15, 15, 15, 15, 15, 15, 17, 17, 17, 17, 20, 20, 21, 21, 21, 21, 22, 22, 22, 22, 22, 22, 24, 24, 24, 24, 24, 24, 25, 25, 27, 27, 27, 27, 27, 27, 28, 28, 28, 28, 29, 29, 32, 32

Offset: 1

Author

Labos Elemer, Jul 11 2000

Keywords

• sign

Comments

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

Examples

			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.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

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

Programs

Mathematica

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

PARI

a(n) = {my(f = n!); valuation(eulerphi(f), 2) - valuation(f, 2);} \\ Amiram Eldar, Jul 15 2024

Python

from math import factorial, prod
from sympy import primerange
from fractions import Fraction
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

Formula

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