A053025 Number of iterations of the number of divisors function (A000005) required to reach a fixed point (1 or 2) when started at n!.

1, 1, 4, 5, 4, 6, 7, 7, 7, 5, 7, 7, 7, 7, 7, 6, 7, 7, 7, 7, 7, 8, 7, 7, 7, 7, 6, 6, 8, 6, 7, 6, 7, 8, 6, 8, 7, 5, 7, 6, 8, 6, 6, 8, 8, 8, 8, 8, 8, 6, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 7, 8, 7, 8, 7, 8, 8, 8, 8, 8, 8, 7, 8, 8, 8, 8, 8, 8, 7, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 7, 8, 7, 8, 8, 8, 7, 8, 8, 7, 8, 8, 8, 8, 8

Offset: 1

Author

Labos Elemer, Feb 24 2000

Keywords

• nonn

Examples

			For n = 108, a(108) = 9 because the sequence of iterates is {108!, 798687560466432000, 7920, 60, 12, 6, 4, 3, 2}, and its length is 9.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000005, A036459, A000142.

Programs

Mathematica

a[n_] := -1 + Length @ FixedPointList[DivisorSigma[0, #] &, n!]; Array[a, 100]  (* Amiram Eldar, Aug 17 2024 *)

PARI

a(n) = {my(f = n!, c = 1); while(f > 2, f = numdiv(f); c++); c;} \\ Amiram Eldar, Aug 17 2024

Formula

a(n) = A036459(A000142(n)) = A036459(n!).