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A127202 a(1)=1, a(2)=2; a(n) = the smallest positive integer not occurring earlier in the sequence such that gcd(a(n), a(n-1)) does not equal gcd(a(n-1), a(n-2)).

%I A127202 #65 Jan 22 2022 23:36:49
%S A127202 1,2,4,3,6,5,10,7,14,8,9,12,11,22,13,26,15,18,16,17,34,19,38,20,21,24,
%T A127202 23,46,25,30,27,28,32,29,58,31,62,33,36,35,40,37,74,39,42,41,82,43,86,
%U A127202 44,45,48,47,94,49,56,50,51,54,52,53,106,55,60,57,59,118,61,122,63,66
%N A127202 a(1)=1, a(2)=2; a(n) = the smallest positive integer not occurring earlier in the sequence such that gcd(a(n), a(n-1)) does not equal gcd(a(n-1), a(n-2)).
%C A127202 This sequence appears to be a permutation of the positive integers. - _Leroy Quet_, Jan 08 2007
%C A127202 From _N. J. A. Sloane_, Jan 26 2017: (Start)
%C A127202 Theorem: This is a permutation of the positive integers.
%C A127202 Proof: (Outline. For details see the link.)
%C A127202 1. Sequence is infinite.
%C A127202 2. For all m, either m is in the sequence or there exists an n_0 such that for n >= n_0, a(n) > m.
%C A127202 3. For all primes p, there is a term divisible by p.
%C A127202 4. For all primes p, there are infinitely many multiples of p in the sequence.
%C A127202 5. Every prime appears in the sequence.
%C A127202 6. For any number m, there are infinitely many multiples of m in the sequence.
%C A127202 7. Every number m appears in the sequence.
%C A127202 (End)
%C A127202 Comment from _N. J. A. Sloane_, Feb 28 2017: (Start)
%C A127202 There are several short cycles and at least one apparently infinite orbit:
%C A127202 [1], [2], [3, 4], [5, 6], [7, 10, 8],
%C A127202 [9, 14, 22, 19, 16, 26, 24, 20, 17, 15, 13, 11],
%C A127202 [21, 34, 29, 25],
%C A127202 and the first apparently infinite orbit is, in the forward direction,
%C A127202 [23, 38, 33, 32, 28, 46, 41, 40, 35, 58, 51, 45, 42, 37, 62, 106, ...] (see A282712), and in the reverse direction
%C A127202 [23, 27, 31, 36, 39, 44, 50, 57, 65, 73, 82, 47, 53, 61, 68, 77, ...] (see A282713). (End)
%C A127202 Conjecture: The two lines in the graph are (apart from small local deviations) defined by the same equations as the two lines in the graph of A283312. - _N. J. A. Sloane_, Mar 12 2017
%H A127202 N. J. A. Sloane, <a href="/A127202/b127202.txt">Table of n, a(n) for n = 1..75000</a> (First 10000 terms from Rémy Sigrist)
%H A127202 Dana G. Korssjoen, Biyao Li, Stefan Steinerberger, Raghavendra Tripathi, and Ruimin Zhang, <a href="https://arxiv.org/abs/2012.04625">Finding structure in sequences of real numbers via graph theory: a problem list</a>, arXiv:2012.04625, Dec 08, 2020
%H A127202 N. J. A. Sloane, <a href="/A127202/a127202_1.txt">Proof that A127202 is a permutation.</a>
%e A127202 gcd(a(7), a(8)) = gcd(10,7) = 1. So a(9) is the smallest positive integer which does not occur earlier in the sequence and which is such that gcd(a(9), 7) is not 1. So a(9) = 14, since gcd(14,7) = 7.
%t A127202 f[l_List] := Block[{k = 1, c = GCD[l[[ -1]], l[[ -2]]]},While[MemberQ[l, k] || GCD[k, l[[ -1]]] == c, k++ ];Append[l, k]];Nest[f, {1, 2}, 69] (* _Ray Chandler_, Jan 16 2007 *)
%o A127202 (PARI) \\ based on Rémy Sigrist's program for A280985
%o A127202 { seen = 0; p = 1; g = 2;
%o A127202         for (n=1, 10000,
%o A127202                 a = 1;
%o A127202 while (bittest(seen, a) || (n>2 && gcd(p,a)==g), a++; );
%o A127202                 print (n " " a);
%o A127202                 g = gcd(p,a);
%o A127202                 p = a;
%o A127202                 seen += 2^a;
%o A127202         )
%o A127202 }
%Y A127202 Cf. A127203, A283312.
%Y A127202 Agrees with A280985 for first 719 terms.
%Y A127202 For fixed points see A281353. See also A282712, A282713.
%K A127202 nonn
%O A127202 1,2
%A A127202 _Leroy Quet_, Jan 08 2007
%E A127202 Extended by _Ray Chandler_, Jan 16 2007