login
A055656
Excess in exponents of powers of 2 in Euler phi of n! compared to that of n!.
1
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
1,5
COMMENTS
The exponents of 2 is larger in phi(n!) than in n! if n > 4.
LINKS
FORMULA
EXAMPLE
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.
MATHEMATICA
eep2[n_]:=Module[{f=n!}, IntegerExponent[EulerPhi[f], 2]-IntegerExponent[f, 2]]; Array[ eep2, 80] (* Harvey P. Dale, Mar 18 2023 *)
PROG
(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
(PARI) a(n) = {my(f = n!); valuation(eulerphi(f), 2) - valuation(f, 2); } \\ Amiram Eldar, Jul 15 2024
CROSSREFS
KEYWORD
sign
AUTHOR
Labos Elemer, Jul 11 2000
STATUS
approved