A055656 Excess in exponents of powers of 2 in EulerPhi 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

1,5

Links

Table of n, a(n) for n=1..74.

Formula

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

Example

n=8, 8!=40320=128*315, Phi(40320)=9216=9*1024 Exponent of 2 in 8! is only 7, in Phi(8!) is 10 so a(8)=10-7=3 Exponents of 2 is larger in Phi(n!) than in n! if n>4.

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

Crossrefs

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

Sequence in context: A241139 A294242 A241092 * A078571 A362970 A194343

Adjacent sequences: A055653 A055654 A055655 * A055657 A055658 A055659

Keyword

sign

Author

Labos Elemer, Jul 11 2000

Status

approved