A055597 Powers of 2 in 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

Table of n, a(n) for n=1..71.

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!.

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

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

Status

approved