A263650 A variation on A098550 (the Yellowstone permutation): a(n)=n for 1 <= n <= 3, a(4)=5; otherwise a(n) = smallest number not yet appearing in the sequence which is coprime to a(n-1) and not coprime to a(n-2).

1, 2, 3, 5, 6, 25, 4, 15, 8, 9, 10, 21, 16, 7, 12, 35, 18, 49, 20, 63, 22, 27, 11, 24, 55, 14, 33, 26, 45, 13, 30, 91, 32, 39, 28, 51, 38, 17, 19, 34, 57, 40, 69, 44, 23, 36, 115, 42, 65, 46, 75, 52, 81, 50, 87, 56, 29, 48, 145, 54, 85, 58, 95, 62, 105, 31, 60, 217, 64

Offset: 1

Author

Bob Selcoe, Oct 22 2015

Keywords

• nonn

Comments

Proof that this is a permutation of the natural numbers follows the same basic format as the proof contained in A098550.

This sequence is one in a multitude of permutations of definable infinite sets (i.e., "infinite permutations") which share similar properties and similar proofs as A098550 (Yellowstone-type), and which are often (though not always - see for example A119718 and A255582) of the general form: a(n) is smallest number not yet appearing in the sequence which is coprime to a(n-1) and not coprime to a(n-2). But caution is warranted here: many sequences which may appear at first glance to be Yellowstone-type infinite permutations are not (e.g., A263648 is infinite, similar in structure to A119718 and even MORE similar to the general Yellowstone form, yet is not a permutation) or may not be provable in similar fashion (e.g., A254077, which is similar in structure to A255582 but cannot be demonstrated as infinite using Yellowstone-type constructions). What observations or generalizations might we draw from this?

Links

• Jean-François Alcover, Table of n, a(n) for n = 1..1000
• David L. Applegate, Hans Havermann, Bob Selcoe, Vladimir Shevelev, N. J. A. Sloane, and Reinhard Zumkeller, The Yellowstone Permutation, arXiv preprint arXiv:1501.01669 [math.NT], 2015.
• David L. Applegate, Hans Havermann, Bob Selcoe, Vladimir Shevelev, N. J. A. Sloane, and Reinhard Zumkeller, The Yellowstone Permutation, Journal of Integer Sequences, Vol. 18 (2015), Article 15.6.7

Crossrefs

Cf. A098550, A119718, A255582, A263648.

Programs

Mathematica

a[n_] := a[n] = If[n <= 4, {1, 2, 3, 5}[[n]], For[k = 4, True, k++, If[CoprimeQ[k, a[n-1]] && !CoprimeQ[k, a[n-2]], If[FreeQ[Array[a, n-1], k], Return[k]]]]]; Array[a, 100] (* Jean-François Alcover, Feb 11 2019 *)

Extensions

Corrected and extended by Jean-François Alcover, Feb 11 2019