A343911 - OEIS

%I #16 Nov 16 2024 03:10:01

%S 0,0,1,1,2,1,2,2,2,2,2,2,3,2,3,3,4,2,3,3,3,2,2,3,3,3,3,3,3,3,3,4,3,4,

%T 4,3,4,3,4,4,4,3,3,3,4,2,2,4,3,3,5,4,3,3,4,4,4,3,2,4,4,3,4,5,5,3,3,5,

%U 3,4,3,4,5,4,4,4,4,4,3,5,4,4,2,4,6,3,4,4,4,4,5,3,4,2

%N a(n) = Omega(phi(n)), where Omega is the number of prime factors of n with multiplicity and phi is the Euler totient function.

%H Amiram Eldar, <a href="/A343911/b343911.txt">Table of n, a(n) for n = 1..10000</a>

%H Paul Erdős and Carl Pomerance, <a href="https://doi.org/10.1216/RMJ-1985-15-2-343">On the normal number of prime factors of phi(n)</a>, Rocky Mountain J. Math., Vol. 15, No. 2 (1985), pp. 343-352.

%F a(n) = A001222(A000010(n)).

%F Limit_{x -> oo} (1/x) * Card({n <= x, a(n) - (1/2)*log(log(x))^2 <= (u/sqrt(3))*log(log(x))^(3/2)}) = (1 + erf(u/sqrt(2)))/2, for every real number u (Erdős and Pomerance, 1985). - _Amiram Eldar_, Nov 16 2024

%t Table[PrimeOmega[EulerPhi[n]], {n, 100}]

%o (PARI) a(n) = bigomega(eulerphi(n)); \\ _Amiram Eldar_, Nov 16 2024

%Y Cf. A000010 (phi), A001222 (Omega).

%K nonn,easy

%O 1,5

%A _Wesley Ivan Hurt_, May 03 2021