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A097465 a(1) = 1; for n>1, a(n) = smallest positive integer which is not among earlier terms of sequence, is coprime to a(n-1) and is not equal to a(n-1) +- 1.

%I A097465 #23 May 06 2022 21:38:36
%S A097465 1,3,5,2,7,4,9,11,6,13,8,15,17,10,19,12,23,14,25,16,21,26,29,18,31,20,
%T A097465 27,22,35,24,37,28,33,38,41,30,43,32,39,34,45,47,36,49,40,51,44,53,42,
%U A097465 55,46,57,50,59,48,61,52,63,58,65,54,67,56,69,62,71,60,73,64,75,68,77
%N A097465 a(1) = 1; for n>1, a(n) = smallest positive integer which is not among earlier terms of sequence, is coprime to a(n-1) and is not equal to a(n-1) +- 1.
%C A097465 A permutation of the positive integers. [This follows easily by standard arguments. One shows that all primes appear. If a number k is missing, find a prime term P that is much greater than k. Then the term following P will be k. Contradiction. - _N. J. A. Sloane_, May 06 2022]
%H A097465 Michael De Vlieger, <a href="/A097465/b097465.txt">Table of n, a(n) for n = 1..10000</a>
%e A097465 a(8) = 11 because, among the positive integers not occurring earlier in the sequence (6,8,10,11,12,...), 11 is the smallest which is coprime to a(7)=9, is not a(7)+1=10 and is not a(7)-1=8.
%p A097465 A:=proc(n) option remember; local t, S; S:=({$1..1000} minus {seq(A(i),i=1..n-1)}) minus {A(n-1)-1,A(n-1)+1}; t:=min(S[]); while igcd(A(n-1),t)>1 do S:=S minus {t}; t:=min(S[]) od; t end: A(1):=1: seq(A(n), n=1..200); # Alec Mihailovs (alec(AT)mihailovs.com), Aug 23 2004
%t A097465 a[1] = 1; a[n_] := a[n] = Block[{t = Table[ a[i], {i, n - 1}], k = 2}, While[k == a[n - 1] - 1 || k == a[n - 1] + 1 || GCD[a[n - 1], k] != 1 || Position[t, k] != {}, k++ ]; k]; Table[ a[n], {n, 50}] (* _Robert G. Wilson v_, Aug 23 2004 *)
%o A097465 (Python)
%o A097465 from math import gcd
%o A097465 from itertools import islice
%o A097465 def agen(): # generator of terms
%o A097465     an, aset, mink = 1, {1}, 2
%o A097465     while True:
%o A097465         yield an
%o A097465         k = mink
%o A097465         while k in aset or gcd(an, k) != 1 or abs(k-an) == 1: k += 1
%o A097465         an = k
%o A097465         aset.add(an)
%o A097465         while mink in aset: mink += 1
%o A097465 print(list(islice(agen(), 72))) # _Michael S. Branicky_, May 02 2022
%Y A097465 Cf. A093714, A097467, A353711.
%K A097465 nonn
%O A097465 1,2
%A A097465 _Leroy Quet_, Aug 23 2004
%E A097465 More terms from Alec Mihailovs (alec(AT)mihailovs.com) and _Robert G. Wilson v_, Aug 23 2004