A384275 a(1) = 1, a(2) = 2, a(3) = 4; for n > 3, a(n) is the smallest unused positive number that shares a factor with a(n-1) and at least one other previous term.

1, 2, 4, 6, 8, 10, 12, 3, 9, 15, 5, 20, 14, 16, 18, 21, 7, 28, 22, 24, 26, 30, 25, 35, 40, 32, 34, 36, 27, 33, 11, 44, 38, 42, 39, 13, 52, 46, 48, 45, 50, 54, 51, 17, 68, 56, 49, 63, 57, 19, 76, 58, 60, 55, 65, 70, 62, 64, 66, 69, 23, 92, 72, 74, 78, 75

Offset

1,2

Comments

For the terms studied, the primes appear in their natural order, and except for a(8) = 3, all other prime terms p are preceded by 3*p and followed by 4*p. The primes p and the terms 4*p form the lower and upper lines of the graph.

In the first 100000 terms the only fixed points are 1, 2, 9; it is likely no more exist. The sequence is conjectured to be a permutation of the positive integers.

An alternate definition of this sequence is 'a(1) = 1, a(2) = 2; for n > 2, a(n) is the smallest unused positive number that shares a factor with a(n-1) while a(n) does not equal a(n-1)/2'. Surprisingly this definition appears to lead to exactly the same sequence. Scott R. Shannon, Mar 01 2026

Links

Scott R. Shannon, Table of n, a(n) for n = 1..10000

Example

a(8) = 3 as a(7) = 12 and a(4) = 6, and 3 shares a factor with 12, the previous term, and also with 6.

Mathematica

nn = 1000; c[_] := False; q = 2; j = 2; c[1] = c[2] = True; u = 3;
  rad[x_] := Times @@ FactorInteger[x][[;; , 1]];
{1, 2}~Join~Reap[Do[k = u;
  While[Or[c[k], CoprimeQ[j, k], CoprimeQ[k, q]], k++];
  Sow[k]; Set[{c[k], q, j}, {True, rad[q*j], k}];
If[k == u, While[c[u], u++]], nn - 2] ][[-1, 1]] (* Michael De Vlieger, May 24 2025 *)

Crossrefs

Cf. A064413, A336957, A098550, A373390.

Sequence in context: A096922 A241071 A393082 * A353026 A356431 A345444

Adjacent sequences: A384272 A384273 A384274 * A384276 A384277 A384278

Keyword

nonn

Author

Scott R. Shannon, May 24 2025

Status

approved