cp's OEIS Frontend

A325410 Smallest k such that the adjusted frequency depth of k! is n > 2.

%I A325410 #6 Apr 25 2019 09:33:16
%S A325410 3,4,5,7,26,65,942,24147
%N A325410 Smallest k such that the adjusted frequency depth of k! is n > 2.
%C A325410 If infinite terms were allowed, we would have a(0) = 1, a(1) = 2, a(2) = infinity. It is possible this sequence is finite, or that there are additional gaps.
%C A325410 The adjusted frequency depth of a positive integer n is 0 if n = 1, and otherwise it is 1 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.
%e A325410 Column n is the sequence of images under A181819 starting with a(n)!:
%e A325410   6  24  120  5040  403291461126605635584000000
%e A325410   4  10  20   84    11264760
%e A325410   3  4   6    12    240
%e A325410      3   4    6     28
%e A325410          3    4     6
%e A325410               3     4
%e A325410                     3
%t A325410 fdadj[n_Integer]:=If[n==1,0,Length[NestWhileList[Times@@Prime/@Last/@FactorInteger[#]&,n,!PrimeQ[#]&]]];
%t A325410 dat=Table[fdadj[n!],{n,1000}];
%t A325410 Table[Position[dat,k][[1,1]],{k,3,Max@@dat}]
%Y A325410 a(n) is the first position of n in A325272.
%Y A325410 Cf. A000142, A022559, A181819, A181821, A323023, A325238, A325273, A325274, A325275, A325276, A325277.
%Y A325410 Omega-sequence statistics: A001222 (first omega), A001221 (second omega), A071625 (third omega), A323022 (fourth omega), A304465 (second-to-last omega), A182850 or A323014 (frequency depth), A325248 (Heinz number), A325249 (sum).
%K A325410 nonn,more
%O A325410 3,1
%A A325410 _Gus Wiseman_, Apr 24 2019