A375311 a(1) = 0; for n >= 2, a(n) is the number of iterations needed for the map: x -> x / k if x is divisible by k, x -> x + 1 otherwise, to (the first occurrence of) 1. Divisor k starts at 2 and its value is increased by 1 after each division.

0, 1, 4, 3, 3, 2, 8, 7, 7, 6, 6, 5, 7, 6, 6, 5, 5, 4, 6, 5, 5, 4, 4, 3, 13, 12, 12, 11, 11, 10, 12, 11, 11, 10, 10, 9, 11, 10, 10, 9, 9, 8, 10, 9, 9, 8, 8, 7, 12, 11, 11, 10, 10, 9, 11, 10, 10, 9, 9, 8, 10, 9, 9, 8, 8, 7, 9, 8, 8, 7, 7, 6, 11, 10, 10, 9, 9, 8

Offset

1,3

Comments

The trajectory length a(n) to (the first occurrence of) 1 is a(n) = r + 1 - n mod 2 + T, where r is the least number such that r! >= n and T is a sum of coefficients c_i in (r! - n - 2 + n mod 2) = c_1*2! + c_2*3! + ... + c_(r-1)*(r-1)!, where 1 <= c_i <= c_(i+1). It resembles the "sum of digits in the factorial base".

Links

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

Formula

a(n!) = n - 1.

Example

n = 9: k = 2,  9 --> 10 --> 5 --> 6 --> 2 --> 3 --> 4 --> 1, thus a(9) = 7.
n = 24: k = 2, 24 --> 12 --> 4 --> 1, thus a(24) = 3.

Mathematica

a[n_] := Module[{x = n, k = 2, c = 0}, While[x > 1, x = If[Divisible[x, k], x/k++, x + 1]; c++]; c]; Array[a, 100] (* Amiram Eldar, Aug 11 2024 *)

Crossrefs

Cf. A000142.

Sequence in context: A016502 A305743 A117691 * A243564 A358329 A171627

Adjacent sequences: A375308 A375309 A375310 * A375312 A375313 A375314

Keyword

nonn

Author

Ctibor O. Zizka, Aug 11 2024

Status

approved