A055597 Exponent of the highest power of 2 dividing phi(n!).

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

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

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

Cf. A000010, A000720, A007814, A011371, A048855.

Sequence in context: A163621 A124194 A359506 * A053048 A154663 A028983

Adjacent sequences: A055594 A055595 A055596 * A055598 A055599 A055600

Keyword

nonn

Author

Labos Elemer, Jul 11 2000

Extensions

Name clarified by Amiram Eldar, Jul 12 2024

Status

approved