A307734 Smallest k such that the adjusted frequency depth of k! is n, and 0 if there is no such k.

1, 2, 0, 3, 4, 5, 7, 26, 65, 942, 24147

Offset: 0

Author

Gus Wiseman, Apr 25 2019

Keywords

• nonn
• more

Comments

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.

Conjecture: this sequence has infinitely many nonzero terms.

Examples

			Column n is the sequence of images under A181819 starting with a(n)!:
  -  2  -  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

Essentially the same as A325410.

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

Cf. A000142, A323023, A325238, A325273, A325274, A325275, A325276, A325277, A325416.

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).