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%I #7 Apr 18 2019 16:55:33
%S 0,1,3,4,5,4,6,6,6,4,6,6,6,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,6,7,
%T 7,7,7,7,6,6,6,6,6,7,7,7,7,7,7,7,6,6,6,6,7,7,7,6,6,6,6,7,7,7,8,7,7,7,
%U 6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7
%N Adjusted frequency depth of n!.
%C The adjusted frequency depth of a positive integer n is 0 if n = 1, and otherwise it is one plus the number of times one must apply A181819 to reach a prime number, where A181819(k = p^i*...*q^j) = prime(i)*...*prime(j) = product of primes indexed by the prime exponents of k. For example, 180 has adjusted frequency depth 5 because we have: 180 -> 18 -> 6 -> 4 -> 3.
%F a(n) = A323014(n!).
%e Recursively applying A181819 starting with 120 gives 120 -> 20 -> 6 -> 4 -> 3, so a(5) = 5.
%t fd[n_]:=Switch[n,1,0,_?PrimeQ,1,_,1+fd[Times@@Prime/@Last/@FactorInteger[n]]];
%t Table[fd[n!],{n,30}]
%Y a(n) = A001222(A325275(n)).
%Y Cf. A000142, A006939, A303555, A323023, A325238, A325273, A325274, A325276, A325277.
%Y Omega-sequence statistics: A001222 (first omega), A001221 (second omega), A071625 (third omega), A323022 (fourth omega), A304465 (second-to-last omega), A182850 or A323014 (length/frequency depth), A325248 (Heinz number), A325249 (sum).
%K nonn
%O 1,3
%A _Gus Wiseman_, Apr 18 2019
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