Voozh

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,3

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 ?

LINKS

Table of n, a(n) for n=1..84.

FORMULA

a(A360806(n)) = n.

EXAMPLE

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.

MATHEMATICA

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

CROSSREFS

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

Sequence in context: A194755 A333884 A055118 * A132671 A340258 A074921