A374540 -id:A374540 - cp's OEIS frontend

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

Ctibor O. Zizka, Aug 01 2024

nonn

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 ?

			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.

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

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

a(A360806(n)) = n.

cp's OEIS Frontend

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.

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