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A247225 a(n) = n if n <= 3, a(4)=5, otherwise the smallest number not occurring earlier having at least one common factor with a(n-3), but none with a(n-1)*a(n-2).

%I A247225 #90 Jul 20 2021 23:48:03
%S A247225 1,2,3,5,4,9,25,8,21,55,16,7,11,6,35,121,12,49,143,10,63,13,20,27,91,
%T A247225 22,15,119,26,33,17,14,39,85,28,57,65,32,19,45,34,133,69,40,77,23,18,
%U A247225 175,253,24,95,161,36,125,203,38,75,29,44,51,145,46,81,155,52
%N A247225 a(n) = n if n <= 3, a(4)=5, otherwise the smallest number not occurring earlier having at least one common factor with a(n-3), but none with a(n-1)*a(n-2).
%C A247225 Conjecturally the sequence is a permutation of the positive integers. However, to prove this we need more subtle arguments than were used to prove the corresponding property for A098550. - _Vladimir Shevelev_, Jan 14 2015
%C A247225 For n <= 2000, a(3n-1) is even and both a(3n) and a(3n-2) are odd numbers. I conjecture that this is true for all positive integers n. This conjecture is true iff for all positive integers n, a(3n-1) is even. - _Farideh Firoozbakht_, Jan 14 2015
%C A247225 From _Vladimir Shevelev_, Jan 19 2015:  (Start)
%C A247225 A generalization of A098550 and A247225.
%C A247225 Let p_n=prime(n). Define the following sequence
%C A247225 a(1)=1, a(2)=p_1,...,a(k+2)=p_(k+1), otherwise the smallest number not occurring earlier having at least one common factor with a(n-(k+1)), but none with a(n-1)*a(n-2)*...*a(n-k).
%C A247225 The sequence begins
%C A247225 1, p_1, p_2, ..., p_(k+1), p_1^2, p_2^2, ..., p_(k+1)^2, p_1^3, ... (*)
%C A247225 [ p_1^3 is followed by p_2*p_(k+2), k<=2,
%C A247225 p_2^3, k>=3, etc.]
%C A247225 In particular, if k=1, it is A098550, if k=2, it is A247225.
%C A247225 Conjecturally for every k>=2, as in the case k=1, the sequence (*) is a permutation of the positive integers. For k>=3, at first glance, already the appearance of the number 6 seems problematic. However, at the author's request, _Peter J. C. Moses_ found that the positions of 6 are 83, 157, 1190, 206, ... in cases k=3,4,5,6,... respectively (A254003).
%C A247225 Note also that for every k>=2, every even term is followed by k odd terms. This is explained by the minimal growth of even numbers (2n) relatively with one of the numbers with the smallest prime divisor p>=3 (asymptotically 6n, 15n, 105n/4, 385n/8, ... for p = 3,5,7,11,... respectively (cf. A084967 - A084970)).
%C A247225 (End)
%H A247225 Peter J. C. Moses, <a href="/A247225/b247225.txt">Table of n, a(n) for n = 1..2000</a>
%H A247225 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, 2015.
%t A247225 a[n_ /; n <= 3] := n; a[4]=5; a[n_] := a[n] = For[aa = Table[a[j], {j, 1, n-1}]; k=4, True, k++, If[FreeQ[aa, k] && !CoprimeQ[k, a[n-3]] && CoprimeQ[k, a[n-1]*a[n-2]], Return[k]]]; Table[ a[n], {n, 1, 65}] (* _Jean-François Alcover_, Jan 12 2015 *)
%Y A247225 Cf. A098550, A247942, A249167, A251604, A254003.
%K A247225 nonn
%O A247225 1,2
%A A247225 _Vladimir Shevelev_, Jan 11 2015
%E A247225 More terms from _Peter J. C. Moses_, Jan 12 2015