A374540 - OEIS

%I #10 Jul 19 2024 21:42:18

%S 0,1,2,1,2,1,1,1,1,1,1,1,3,1,1,3,1,2,1,1,1,1,1,1,1,1,1,1,1,3,1,1,1,1,

%T 1,1,1,1,1,1,1,2,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,1,1,

%U 1,1,1,2,1,1,1,1,1,1,1,1,1,2,1,1,1,1,2,1,1,1,1,1,1,1,1,1,1

%N a(1) = 0; for n >= 2, a(n) is the number of iterations needed for the map x -> x/A000005(x) to reach a least integer, when starting from x = A033950(n).

%C The refactorability "depth" for refactorable numbers. Numbers from A159973 have the refactorability "depth" 0. Records reached for A033950(A360806(n)), i.e. the growth of the sequence is very slow.

%e n = 2: A033950(2) = 2, 2/A000005(2) = 1, thus a(2) = 1.

%e n = 3: A033950(3) = 8, 8/A000005(8) = 2 --> 2/A000005(2) = 1, thus a(3) = 2.

%e n = 13: A033950(13) = 80, 80/A000005(80) = 8 --> 8/A000005(8) = 2 --> 2/A000005(2) = 1, thus a(13) = 3.

%t f[n_] := Module[{v = NestWhileList[# / DivisorSigma[0, #] &, n, IntegerQ[#] && # > 1 &], len}, len = Length[v]; If[IntegerQ[v[[2]]], If[v[[-1]] == 1, len - 1, len - 2], Nothing]]; f[1] = 0; Array[f, 1200] (* _Amiram Eldar_, Jul 11 2024 *)

%Y Cf. A000005, A033950, A036762, A159973, A360806.

%K nonn

%O 1,3

%A _Ctibor O. Zizka_, Jul 11 2024