A377403 For n >= 2, a(n) is the number of iterations needed for the map: x -> x / A085392(x) if A085392(x) > 1, otherwise x -> x + A151800(x), to (the first occurrence of) 2.

0, 3, 1, 3, 1, 3, 2, 4, 1, 4, 2, 3, 1, 4, 3, 4, 2, 3, 2, 4, 1, 3, 3, 4, 1, 5, 2, 4, 2, 3, 4, 4, 1, 4, 3, 3, 1, 4, 3, 4, 2, 4, 2, 5, 1, 4, 4, 4, 2, 4, 2, 5, 3, 4, 3, 4, 1, 5, 3, 7, 1, 5, 5, 4, 2, 3, 2, 4, 2, 6, 4, 4, 1, 5, 2, 4, 2, 5, 4, 6, 1, 3, 3, 4, 1, 4, 3, 3, 3, 4, 2, 4, 1, 4, 5, 4, 2, 5, 3, 4, 2, 4, 3, 5, 1, 6, 4, 3, 2, 4, 4, 6, 2, 4, 2, 5, 1, 4, 4, 5

Offset: 2

Author

Ctibor O. Zizka, Oct 27 2024

Keywords

• nonn

Comments

Also a(2*k + 1) = A001222(2*k + 1) + 2 + s, where s >= 1 for k = 5, 8, 14, 20, 21, 23, 26, 29, 30, 35, 36, 39, 48, 50, 51, ...

Examples

			n = 3: 3 -> 8 -> 4 -> 2, 3 iterations needed to reach 2, thus a(3) = 3.
n = 9: 9 -> 3 -> 8 -> 4 -> 2, 4 iterations needed to reach 2, thus a(9) = 4.
n = 11: 11 -> 24 -> 8 -> 4 - > 2, 4 iterations needed to reach 2, thus a(11) = 4.
		

Crossrefs

Cf. A001222, A013634, A020639, A085392, A107286, A151800.

Programs

Mathematica

a[2] = 0; a[n_] := -1 + Length@ NestWhileList[If[CompositeQ[#], #/FactorInteger[#][[-1, 1]], # + NextPrime[#]] &, n, # > 2 &]; Array[a, 120, 2] (* Amiram Eldar, Oct 27 2024 *)

Formula

For n even: a(n) = A001222(n) - 1.

For n odd: a(n) = A001222(n) - 1 + A001222(A013634(A020639(n))).