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%I A263650 #19 Feb 11 2019 08:44:50
%S A263650 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,
%T A263650 33,26,45,13,30,91,32,39,28,51,38,17,19,34,57,40,69,44,23,36,115,42,
%U A263650 65,46,75,52,81,50,87,56,29,48,145,54,85,58,95,62,105,31,60,217,64
%N 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).
%C A263650 Proof that this is a permutation of the natural numbers follows the same basic format as the proof contained in A098550.
%C A263650 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?
%H A263650 Jean-François Alcover, <a href="/A263650/b263650.txt">Table of n, a(n) for n = 1..1000</a>
%H A263650 David L. Applegate, Hans Havermann, Bob Selcoe, Vladimir Shevelev, N. J. A. Sloane, and Reinhard Zumkeller, <a href="http://arxiv.org/abs/1501.01669">The Yellowstone Permutation</a>, arXiv preprint arXiv:1501.01669 [math.NT], 2015.
%H A263650 David L. Applegate, Hans Havermann, Bob Selcoe, Vladimir Shevelev, N. J. A. Sloane, and Reinhard Zumkeller, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL18/Sloane/sloane9.html">The Yellowstone Permutation</a>, Journal of Integer Sequences, Vol. 18 (2015), Article 15.6.7
%t A263650 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 *)
%Y A263650 Cf. A098550, A119718, A255582, A263648.
%K A263650 nonn
%O A263650 1,2
%A A263650 _Bob Selcoe_, Oct 22 2015
%E A263650 Corrected and extended by _Jean-François Alcover_, Feb 11 2019

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