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A352950 Lexicographically earliest infinite sequence of distinct nonnegative integers commencing 1,3,5,7 such that any four consecutive terms are pairwise coprime.

%I A352950 #24 Jul 01 2025 23:33:33
%S A352950 1,3,5,7,2,9,11,13,4,15,17,19,8,21,23,25,16,27,29,31,10,33,37,41,14,
%T A352950 39,43,47,20,49,51,53,22,35,57,59,26,55,61,63,32,65,67,69,28,71,73,45,
%U A352950 34,77,79,75,38,83,89,81,40,91,97,87,44,85,101,93,46,95,103,99,52,107,109,105
%N A352950 Lexicographically earliest infinite sequence of distinct nonnegative integers commencing 1,3,5,7 such that any four consecutive terms are pairwise coprime.
%C A352950 The pairwise coprime relations found in the first four odd numbers 1,3,5,7 are preserved throughout in any run of four consecutive terms.
%C A352950 a(4n+5) is always even (and < a(4n+2)); n>=0.
%C A352950 The plot exhibits two distinct rays at first (upper/odd, lower/even), with no terms divisible by 6 until a(229), at which point the even ray switches to producing just 28 multiples of 6 until a(337)=168. At this point the original even ray is re-established, the odd ray divides into two (quasi-parallel) rays, and no further multiples of 6 are seen. Therefore it seems very unlikely that the sequence is a permutation of the nonnegative integers.
%C A352950 Primes p other than p = 2 appear in their natural order.
%H A352950 Michael De Vlieger, <a href="/A352950/a352950.png">Log-log scatterplot of a(n)</a>, n = 1..2^14, highlighting primes in green, numbers divisible by 6 in gold, other even numbers in red, odd numbers divisible by 3 in blue.
%e A352950 3,5,7 are pairwise coprime and 2 is the smallest unused number coprime to all of them, therefore a(5)=2.
%p A352950 ina := proc(n) false end: # adapted from code for A103683
%p A352950 a := proc (n) option remember; local k;
%p A352950 if n < 5 then k := 2*n-1
%p A352950 else for k from 2 while ina(k) or igcd(k, a(n-1)) <> 1 or igcd(k, a(n-2)) <> 1 or igcd(k, a(n-3)) <> 1
%p A352950 do
%p A352950 end do
%p A352950 end if; ina(k):= true; k
%p A352950 end proc;
%p A352950 seq(a(n), n = 1 .. 100);
%t A352950 Block[{a, c, k, m, u, nn}, nn = 86; c[_] = 0; MapIndexed[Set[{a[First[#2]], c[#1]}, {#1, First[#2]}] &, {1, 3, 5, 7}]; u = 2; Do[k = u; m = LCM @@ Array[a[i - #] &, 3]; While[Nand[c[k] == 0, CoprimeQ[m, k]], k++]; Set[{a[i], c[k]}, {k, i}]; If[a[i] == u, While[c[u] > 0, u++]], {i, 5, nn}]; Array[a, nn] ] (* _Michael De Vlieger_, Apr 12 2022 *)
%o A352950 (Python)
%o A352950 from math import gcd
%o A352950 from itertools import islice
%o A352950 def agen(): # generator of terms
%o A352950     aset, b, c, d = {1, 3, 5, 7}, 3, 5, 7
%o A352950     yield from [1, b, c, d]
%o A352950     while True:
%o A352950         k = 1
%o A352950         while k in aset or any(gcd(t, k) != 1 for t in [b, c, d]): k+= 1
%o A352950         b, c, d = c, d, k
%o A352950         aset.add(k)
%o A352950         yield k
%o A352950 print(list(islice(agen(), 107))) # _Michael S. Branicky_, Apr 10 2022
%Y A352950 Cf. A085229, A103683, A350359.
%K A352950 nonn
%O A352950 1,2
%A A352950 _David James Sycamore_, Apr 10 2022