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%I A280985 #38 Nov 03 2020 02:44:32
%S A280985 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 A280985 23,46,25,30,27,28,32,29,58,31,62,33,36,35,40,37,74,39,42,41,82,43,86,
%U A280985 44,45,48,47,94,49,56,50,51,54,52,53,106,55,60,57,59,118
%N A280985 a(1)=1, and then a(n) = smallest positive integer not occurring earlier in the sequence sharing some prime factor with at least one of a(n-1) and a(n+1).
%C A280985 In other words, for any n>1, gcd(a(n), a(n-1))*gcd(a(n), a(n+1)) > 1.
%C A280985 This sequence is related to A127202: here we require that the derived sequence b(n) = gcd(a(n), a(n+1)) does not contain two consecutive ones, there we require that the derived sequence c(n) = gcd(A127202(n), A127202(n+1)) (see A127203) does not contain two consecutive equal values; this sequence first differs from A127202 at n=720: a(720)=666 whereas A127202(720)=667.
%C A280985 This sequence is also related to the EKG sequence (A064413): here we require a common prime factor with at least one neighbor, there we require a common prime factor with both neighbors.
%C A280985 This sequence is a permutation of the natural numbers, with inverse A281117: Proof:
%C A280985 - The sequence is injective by definition,
%C A280985 - The sequence is surjective: by contradiction: let m be the least value missing from the sequence, and n0 the least value such that a(n)>m for any n>=n0; if a(n0) shares a prime factor with a(n0-1), then we can choose a(n0+1)=m; if a(n0) does not share a prime factor with a(n0-1), then a(n0+1) shares a prime factor with a(n0), and we can choose a(n0+2)=m: contradiction. QED
%H A280985 N. J. A. Sloane, Table of n, a(n) for n = 1..75000 (first 10000 terms from Rémy Sigrist) Computed using Rémy Sigrist's PARI program.
%H A280985 Rémy Sigrist, PARI program for A280985
%H A280985 Index entries for sequences that are permutations of the natural numbers
%F A280985 The following heuristic argument explains why the graph of this sequence is almost identical to the graph of A283312. Numbers appear in the sequence in their natural order, except when interrupted either by the appearance of a prime p, which is always followed by 2*p, or by a composite number c which needs to be followed by a missing number c' with gcd(c, c') > 1 (a(29) = 25 = c with a(30) = 30 = c' is an example). But c' is normally close to c, so those displacements have only a small local effect compared with the larger displacements caused by the primes. So the two lines in the graph are essentially defined by the same equations as the two lines in the graph of A283312. - _N. J. A. Sloane_, Nov 03 2020
%e A280985 The first terms, alongside the GCD with the next term, are:
%e A280985 n a(n) GCD
%e A280985 - ---- ---
%e A280985 1 1 1
%e A280985 2 2 2
%e A280985 3 4 1
%e A280985 4 3 3
%e A280985 5 6 1
%e A280985 6 5 5
%e A280985 7 10 1
%e A280985 8 7 7
%e A280985 9 14 2
%e A280985 10 8 1
%e A280985 11 9 3
%e A280985 12 12 1
%e A280985 13 11 11
%e A280985 14 22 1
%e A280985 ... ... ...
%e A280985 717 661 661
%e A280985 718 1322 2
%e A280985 719 664 2
%e A280985 720 666 1
%e A280985 721 667 23
%t A280985 f[s_List] := Block[{g = GCD[s[[-2]], s[[-1]]], k = 3}, While[ MemberQ[s, k] || GCD[s[[-1]], k] == g, k++]; Append[s, k]]; Nest[f, {1, 2}, 65] (* _Robert G. Wilson v_, Mar 03 2017 *)
%Y A280985 Cf. A064413, A127202 (agrees for first 719 terms), A127203, A281117, A283312.
%Y A280985 For fixed points see A281353, for indices of primes see A338352.
%K A280985 nonn
%O A280985 1,2
%A A280985 _Rémy Sigrist_, Jan 12 2017
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