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A055597
Exponent of the highest power of 2 dividing phi(n!).
1
0, 0, 1, 3, 5, 6, 7, 10, 10, 11, 12, 14, 16, 17, 17, 21, 25, 26, 27, 29, 29, 30, 31, 34, 34, 35, 35, 37, 39, 40, 41, 46, 46, 47, 47, 49, 51, 52, 52, 55, 58, 59, 60, 62, 62, 63, 64, 68, 68, 69, 69, 71, 73, 74, 74, 77, 77, 78, 79, 81, 83, 84, 84, 90, 90, 91, 92, 94, 94, 95, 96
OFFSET
1,4
LINKS
FORMULA
a(n) = A007814(A048855(n)) = A007814(A000010(n!)).
EXAMPLE
For n=8, 8! = 40320 = 128*315, phi(40320) = 9216 = 9*1024, so a(8)=10, while the exponent of 2 in 8! is only 7. Exponents of 2 are larger in phi(n!) than in n!.
MATHEMATICA
a[n_] := IntegerExponent[EulerPhi[n!], 2]; Array[a, 100] (* Amiram Eldar, Jul 12 2024 *)
PROG
(Python)
from math import factorial, prod
from sympy import primerange
from fractions import Fraction
def A055597(n): return (~(m:=(factorial(n)*prod(Fraction(p-1, p) for p in primerange(n+1))).numerator)&m-1).bit_length() # Chai Wah Wu, Jul 06 2022
(PARI) a(n) = valuation(eulerphi(n!), 2); \\ Amiram Eldar, Jul 12 2024
CROSSREFS
KEYWORD
nonn
AUTHOR
Labos Elemer, Jul 11 2000
EXTENSIONS
Name clarified by Amiram Eldar, Jul 12 2024
STATUS
approved