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A357614 Lexicographically earliest infinite sequence of distinct positive integers such that a(n+1) is the least k != j, for which gcd(k, j) > 1, where j = a(n) + prime(n).

%I A357614 #13 Jun 29 2025 16:46:58
%S A357614 1,6,3,2,12,46,118,5,4,9,8,13,10,15,14,122,7,11,16,166,18,21,20,206,
%T A357614 25,22,24,254,19,26,278,27,28,30,39,32,33,34,394,17,35,36,31,37,23,38,
%U A357614 42,44,45,40,538,48,41,47,50,614,1754,49,52,56,674,29,54,57,58
%N A357614 Lexicographically earliest infinite sequence of distinct positive integers such that a(n+1) is the least k != j, for which gcd(k, j) > 1, where j = a(n) + prime(n).
%C A357614 If a(n) + prime(n) is a prime p, then a(n+1) is the smallest multiple (>1) of p that has not occurred earlier. Conjectured to be a permutation of the positive integers.
%H A357614 Michael De Vlieger, <a href="/A357614/b357614.txt">Table of n, a(n) for n = 1..10000</a>
%H A357614 Michael De Vlieger, <a href="/A357614/a357614.png">Log log scatterplot of a(n)</a>, n = 1..2^14, labeling maxima in red and local minima in blue, highlighting primes in green and other prime powers in gold.
%e A357614 a(1)=1; 1 + prime(1)=3, so a(2) = 6, the smallest unused number sharing a divisor with 3.
%t A357614 nn = 65; c[_] = False; u = 2; a[1] = j = 1; c[1] = True; Do[m = j + Prime[n - 1]; If[PrimeQ[m], k = 2; While[c[k m], k++]; k *= m, k = u; While[Or[c[k], k == m, CoprimeQ[k, m]], k++]]; Set[{a[n], c[k], j}, {k, True, k}]; If[k == u, While[c[u], u++]], {n, 2, nn}]; Array[a, nn] (* _Michael De Vlieger_, Oct 06 2022 *)
%Y A357614 Cf. A347113, A349472, A357595.
%K A357614 nonn
%O A357614 1,2
%A A357614 _David James Sycamore_, Oct 06 2022