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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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%I #16 Feb 21 2025 03:50:25

%S 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,

%T 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,

%U 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

%N 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.

%C 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 ?

%F a(A360806(n)) = n.

%e 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.

%e 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.

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

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

%K nonn,changed

%O 1,3

%A _Ctibor O. Zizka_, Aug 01 2024