A325272 Adjusted frequency depth of n!.

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, 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, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7

Offset

1,3

Comments

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.

Links

Table of n, a(n) for n=1..87.

Formula

a(n) = A323014(n!).

Example

Recursively applying A181819 starting with 120 gives 120 -> 20 -> 6 -> 4 -> 3, so a(5) = 5.

Mathematica

fd[n_]:=Switch[n, 1, 0, _?PrimeQ, 1, _, 1+fd[Times@@Prime/@Last/@FactorInteger[n]]];
Table[fd[n!], {n, 30}]

Crossrefs

a(n) = A001222(A325275(n)).

Cf. A000142, A006939, A303555, A323023, A325238, A325273, A325274, 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 (length/frequency depth), A325248 (Heinz number), A325249 (sum).

Sequence in context: A114545 A309704 A232702 * A178698 A225072 A262906

Adjacent sequences: A325269 A325270 A325271 * A325273 A325274 A325275

Keyword

nonn

Author

Gus Wiseman, Apr 18 2019

Status

approved