A375147 a(1) = 0; for n >= 2, a(n) is the number of iterations needed for the map: x -> x / A000005(x) if x is divisible by A000005(x), x -> x + 1 otherwise, to reach 1.

0, 1, 7, 6, 5, 4, 3, 2, 8, 4, 3, 2, 13, 12, 11, 10, 9, 8, 13, 12, 11, 10, 9, 8, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 9, 8, 7, 6, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 9, 8, 7, 6, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 10, 9, 8, 7, 6, 5, 4, 3, 7, 6, 5, 4

Offset: 1

Author

Ctibor O. Zizka, Aug 01 2024

Keywords

• nonn

Comments

The trajectory length is a repeated sum of steps up to the next refactorable number (A360778) and its refactoring "depth" (A374540). The sequence always reach 1 as soon as an iterate reaches the value x from A330816. Assuming A330816 to be finite (conjectured by David A. Corneth) and A360806 to be infinite, may there be a set of numbers n > 10^42, which is not reaching 1 ?

Examples

			x = 3:  the trajectory is 3 --> 4 --> 5 --> 6 --> 7 --> 8 --> 2 --> 1, number of steps needed to reach 1 is 7, thus a(3) = 7.
x = 81: the trajectory is 81 --> 82 --> 83 --> 84 --> 7 --> 8 --> 2 --> 1, number of steps needed to reach 1 is 7, thus a(81) = 7.
		

Crossrefs

Cf. A000005, A033950, A330816, A360778, A360806, A374540.

Programs

Mathematica

a[n_] := -1 + Length[NestWhileList[If[IntegerQ[(r = #/DivisorSigma[0, #])], r, # + 1] &, n, # > 1 &]]; Array[a, 100] (* Amiram Eldar, Aug 01 2024 *)

Formula

a(A360806(n)) = n.