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).

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, 23, 46, 25, 30, 27, 28, 32, 29, 58, 31, 62, 33, 36, 35, 40, 37, 74, 39, 42, 41, 82, 43, 86, 44, 45, 48, 47, 94, 49, 56, 50, 51, 54, 52, 53, 106, 55, 60, 57, 59, 118

Offset: 1

Author

Rémy Sigrist, Jan 12 2017

Keywords

• nonn

Comments

In other words, for any n>1, gcd(a(n), a(n-1))*gcd(a(n), a(n+1)) > 1.

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.

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.

This sequence is a permutation of the natural numbers, with inverse A281117: Proof:

- The sequence is injective by definition,

- 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

Examples

			The first terms, alongside the GCD with the next term, are:
n     a(n)     GCD
-     ----     ---
1        1       1
2        2       2
3        4       1
4        3       3
5        6       1
6        5       5
7       10       1
8        7       7
9       14       2
10       8       1
11       9       3
12      12       1
13      11      11
14      22       1
...    ...     ...
717    661     661
718   1322       2
719    664       2
720    666       1
721    667      23
		

Links

• 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.
• Rémy Sigrist, PARI program for A280985
• Index entries for sequences that are permutations of the natural numbers

Crossrefs

Cf. A064413, A127202 (agrees for first 719 terms), A127203, A281117, A283312.

For fixed points see A281353, for indices of primes see A338352.

Programs

Mathematica

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 *)

Formula

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