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A353730 a(1)=2; thereafter a(n) is the smallest positive number not yet used which is compatible with the condition that a(n) is relatively prime to the next n terms.

%I A353730 #45 May 21 2022 06:36:13
%S A353730 2,1,3,4,5,7,9,11,8,13,17,19,23,25,21,29,31,37,16,41,43,47,53,59,61,
%T A353730 67,71,73,55,79,27,49,83,89,97,101,103,107,26,109,113,127,131,137,139,
%U A353730 149,151,157,163,167,173,179,181,191,193,197,199,211,85,121,223
%N A353730 a(1)=2; thereafter a(n) is the smallest positive number not yet used which is compatible with the condition that a(n) is relatively prime to the next n terms.
%C A353730 Similar to A247665, which is obtained if the condition "smallest positive number" is changed to "smallest number >= 2".
%C A353730 It would be nice to have a proof that the numbers 6, 10, 12, 14, 15, 18, 20, 22, ... are missing from this sequence. It appears that the missing numbers are 6, 10, 12, 14, 15, 18, 20, 22, 24, 28, 30, 33, 34, 35, 36, 38, 39, 40, 42, 44, ..., but since there is no proof that any one of these is really missing, this sequence cannot yet be added to the OEIS.
%H A353730 Russ Cox, <a href="/A353730/b353730.txt">Table of n, a(n) for n = 1..100000</a> (terms 1..1000 from Alois P. Heinz; terms 1..10000 from Chai Wah Wu)
%e A353730 a(1) = 2 must be rel. prime to a(2), so a(2)=1.
%e A353730 a(2) = 1 must be rel. prime to a(3) and a(4), so we can take them to be 3 and 4.
%e A353730 a(3) = 3 must be rel. prime to a(5), a(6), so we can take them to be 5 and 7.
%e A353730 a(4) = 4 must be rel. prime to a(7), a(8), so we can take them to be 9 and 11.
%e A353730 At each step after the first, we must choose two new numbers, and we must make sure that not only are they rel. prime to a(n), they are also rel. prime to all a(i), i>n, that have been already chosen.
%o A353730 (Python)
%o A353730 from itertools import count, islice
%o A353730 from math import gcd
%o A353730 from collections import deque
%o A353730 def A353730_gen(): # generator of terms
%o A353730     aset, aqueue, c, f = {2}, deque([2]), 1, True
%o A353730     yield 2
%o A353730     while True:
%o A353730         for m in count(c):
%o A353730             if m not in aset and all(gcd(m,a) == 1 for a in aqueue):
%o A353730                 yield m
%o A353730                 aset.add(m)
%o A353730                 aqueue.append(m)
%o A353730                 if f: aqueue.popleft()
%o A353730                 f = not f
%o A353730                 while c in aset:
%o A353730                     c += 1
%o A353730                 break
%o A353730 A353730_list = list(islice(A353730_gen(),30)) # _Chai Wah Wu_, May 18-19 2022
%Y A353730 Cf. A247665; A353734 (powers of 2).
%Y A353730 For the even terms, see A354146.
%K A353730 nonn
%O A353730 1,1
%A A353730 _N. J. A. Sloane_, May 16 2022