A330816 -id:A330816 - cp's OEIS frontend

A334814 Least number that reaches 1 after n iterations of the map k -> sigma(k)/d(k) if d(k) | sigma(k), and k -> 1 otherwise, where d(k) is the number of divisors of k (A000005) and sigma(k) is their sum (A000203).

1, 2, 3, 5, 11, 29, 107, 257, 941, 2017, 11261, 45039, 441073, 2151073, 8575873, 42884161, 220268161, 440536321

Offset: 0

Amiram Eldar, May 12 2020

Apparently, most of the terms are primes. 45039 = 3 * 15013 is the first composite term.

a(18) > 2*10^10, if it exists.

			a(3) = 5 since sigma(5)/d(5) = 6/2 = 3, sigma(3)/d(3) = 4/2 = 2, and sigma(2)/d(2) = 3/2 is not an integer, hence there are 3 iterations: 5 -> 3 -> 2 -> 1, and 5 is the least number with 3 iterations.

Cf. A000005, A000203, A003601, A102187, A330816, A334813.

rat[n_] := If[IntegerQ[r = DivisorSigma[1, n]/DivisorSigma[0, n]], r, 1]; f[n_] := Length @ FixedPointList[rat, n] - 1; max = 10; seq = Table[0, {max}]; c = 0; n = 1; While[c < max, i = f[n]; If[i <= max && seq[[i]] == 0, c++; seq[[i]] = n]; n++]; seq

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.

Original entry on oeis.org

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

A334814 Least number that reaches 1 after n iterations of the map k -> sigma(k)/d(k) if d(k) | sigma(k), and k -> 1 otherwise, where d(k) is the number of divisors of k (A000005) and sigma(k) is their sum (A000203).

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