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Exponent of the highest power of 2 dividing phi(n!).
1

%I #17 Jul 12 2024 10:16:58

%S 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,

%T 35,35,37,39,40,41,46,46,47,47,49,51,52,52,55,58,59,60,62,62,63,64,68,

%U 68,69,69,71,73,74,74,77,77,78,79,81,83,84,84,90,90,91,92,94,94,95,96

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

%H Amiram Eldar, <a href="/A055597/b055597.txt">Table of n, a(n) for n = 1..10000</a>

%F a(n) = A007814(A048855(n)) = A007814(A000010(n!)).

%e 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!.

%t a[n_] := IntegerExponent[EulerPhi[n!], 2]; Array[a, 100] (* _Amiram Eldar_, Jul 12 2024 *)

%o (Python)

%o from math import factorial, prod

%o from sympy import primerange

%o from fractions import Fraction

%o 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

%o (PARI) a(n) = valuation(eulerphi(n!), 2); \\ _Amiram Eldar_, Jul 12 2024

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

%K nonn

%O 1,4

%A _Labos Elemer_, Jul 11 2000

%E Name clarified by _Amiram Eldar_, Jul 12 2024