A377027 a(1) = 1. For n > 1, if a(n-1) is a novel term, a(n) = a(a(n-1)), and if a(n-1) seen k (>1) times (including a(n-1)), with k*a(n-1) < n then a(n) = k*a(n-1); else a(n) = 1.

1, 1, 2, 1, 3, 2, 4, 1, 4, 8, 1, 5, 3, 6, 2, 6, 12, 5, 10, 8, 16, 6, 18, 5, 15, 2, 8, 24, 5, 20, 8, 1, 6, 24, 1, 7, 4, 12, 24, 1, 8, 40, 1, 9, 4, 16, 32, 1, 10, 20, 40, 1, 11, 1, 12, 36, 7, 14, 6, 30, 20, 60, 30, 60, 1, 13, 3, 9, 18, 36, 1, 14, 28, 24, 1, 15, 30

Offset

1,3

Comments

The clause requiring a(n) = 1 if k*a(n-1) >= n is necessary to ensure continuation of the sequence, since without it there would be occasions of a(m) = a(a(m)) where a(m) is not yet defined, see Example. This is conjectured to occur infinitely many times as the sequence extends, which then implies that every positive integer will eventually enter the sequence (by incrementation of 1 if not before). Consequently every integer is conjectured to occur infinitely many times.

Links

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Michael De Vlieger, Log log scatterplot of a(n), n = 1..2^20.

Example

a(1) = 1 is a novel term so a(2) = a(a(1)) = a(1) = 1. Then a(3) = 2*1 = 2 because 1 has occurred twice. a(4) = a(a(2)) = a(2) = 1, and so on.
The first occasion of invoking the "a(n) = 1" clause is following a(32) = 32, a novel term so then a(33) = 32, whereupon (without the clause) we would have require a(34) = 64 then a(35) = a(a(64)), where a(64) is not yet defined. To avoid this we set a(34) = 1 so that the sequence may continue.

Mathematica

nn = 2^20; c[_] := 0; a[1] = 1; Do[j = a[n - 1]; If[# == 0, k = a[j], If[# < n, k = #, k = 1] &[(# + 1)*j] ] &[c[j]]; a[n] = k; c[j]++, {n, 2, nn}]; Array[a, nn] (* Michael De Vlieger, Jun 30 2025 *)

Crossrefs

Cf. A376908.

Sequence in context: A003960 A243499 A124223 * A379730 A295908 A300980

Adjacent sequences: A377024 A377025 A377026 * A377028 A377029 A377030

Keyword

nonn,easy

Author

David James Sycamore, Oct 13 2024

Extensions

Corrected and extended by Michael De Vlieger, Jul 01 2025

Status

approved