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

3, 4, 5, 7, 26, 65, 942, 24147

Offset: 3

Author

Gus Wiseman, Apr 24 2019

Keywords

• nonn
• more

Comments

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.

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.

Examples

			Column n is the sequence of images under A181819 starting with a(n)!:
  6  24  120  5040  403291461126605635584000000
  4  10  20   84    11264760
  3  4   6    12    240
     3   4    6     28
         3    4     6
              3     4
                    3
		

Crossrefs

a(n) is the first position of n in A325272.

Cf. A000142, A022559, A181819, A181821, A323023, A325238, A325273, A325274, A325275, A325276, A325277.

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).

Programs

Mathematica

fdadj[n_Integer]:=If[n==1,0,Length[NestWhileList[Times@@Prime/@Last/@FactorInteger[#]&,n,!PrimeQ[#]&]]];
dat=Table[fdadj[n!],{n,1000}];
Table[Position[dat,k][[1,1]],{k,3,Max@@dat}]