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A357963 a(1)=1, a(2)=2. Thereafter, if there are prime divisors p of a(n-1) which do not divide a(n-2), a(n) is the least novel multiple of any such p. Otherwise a(n) is the least novel multiple of the squarefree kernel of a(n-1). See comments.

%I A357963 #17 Jul 03 2025 23:41:10
%S A357963 1,2,4,6,3,9,12,8,10,5,15,18,14,7,21,24,16,20,25,30,22,11,33,27,36,26,
%T A357963 13,39,42,28,56,70,35,105,45,60,32,34,17,51,48,38,19,57,54,40,50,80,
%U A357963 90,63,49,77,44,46,23,69,66,52,65,55,88,58,29,87,72,62,31,93
%N A357963 a(1)=1, a(2)=2. Thereafter, if there are prime divisors p of a(n-1) which do not divide a(n-2), a(n) is the least novel multiple of any such p. Otherwise a(n) is the least novel multiple of the squarefree kernel of a(n-1). See comments.
%C A357963 In other words, if a(n-1) has k prime divisors p_j, 1 <= j <= k which do not divide a(n-2), where 1 <= k <= omega(a(n-1)), and if m_j*p_j is the least multiple of p_j which is not already a term, then a(n) = Min{m_j*p_j; 1<=j<=k}. Otherwise every prime divisor of a(n-1) also divides a(n-2), in which case a(n) is the least multiple of the squarefree kernel of a(n-1) which is not already a term. If a(n-1) and a(n-2) are coprime the computation of a(n) ranges over all prime divisors of a(n-1). This happens only once (n=3), after which all adjacent terms share a common divisor (as in the EKG sequence, A064413).
%C A357963 Departs from A064413 and A352187 at a(19), a(31) respectively, and apparently shares the way odd primes are proven to appear in the former and conjectured to appear in the latter; namely as 2*p, p, 3*p.
%C A357963 Conjectures: Permutation of the positive integers with primes in natural order, appearing in same way as in EKG.
%C A357963 From _Michael De Vlieger_, Oct 22 2022 (Start)
%C A357963 An algorithm similar to the Rains algorithm for the EKG sequence efficiently generates the sequence.
%C A357963 Like the EKG sequence, this sequence forces primes into divisibility; Primes divide their predecessors and successors. Consequently they exhibit Lagarias-Rains-Sloane chain 2p -> p -> 3p outside of p = 2, just as in the EKG sequence.
%C A357963 Let us define several quasi-rays conspicuous in the scatterplot. From lowest to highest apparent slope, we have the following:
%C A357963 - beta: local minima, i.e., a(1)=1 and primes p in order.
%C A357963 - gamma: 2p, 4p, and certain other composites.
%C A357963 - alpha-k: k*p from large k to k = 3. This system appears as a series of fine quasi-rays, with 3p generally comprising records.
%C A357963 Records are 3p outside of {1, 2, 4, 12, 18, 24, 25, 30, 36, 42, 56, 70, 105}.
%C A357963 a(33) = 35 behaves like a prime; 70 precedes and 105 follows it. a(34) = 105 is conspicuous as it appears earlier than expected. (End)
%H A357963 Michael De Vlieger, <a href="/A357963/b357963.txt">Table of n, a(n) for n = 1..10000</a>
%H A357963 Michael De Vlieger, <a href="/A357963/a357963.png">Scatterplot of a(n)</a>, n = 1..120, highlighting primes p in red, 2p in blue, and 3p in green.
%H A357963 Michael De Vlieger, <a href="/A357963/a357963_1.png">Log-log scatterplot of a(n)</a>, n = 1..2^12, labeling the first 64 terms, showing records in red and local minima in blue, highlighting primes in green and other prime powers in gold. As n increases, the quasi-ray 5*p shows prominently just under the records.
%H A357963 Michael De Vlieger, <a href="/A357963/a357963_2.png">Log-log scatterplot of a(n)</a>, n = 1..2^20, demonstrating fine quasi-rays.
%e A357963 a(1)=1, a(2)=2 and 2 divides 2 but does not divide 1. Since 2 is the only prime divisor of 2, a(3) = 4, the least unused multiple of 2.
%e A357963 Since every prime divisor of a(3)=4 also divides a(2)=2, a(4) = 6, the least novel multiple of the squarefree kernel of 4.
%e A357963 a(19), a(20)=25, 30, and 30 has two prime divisors (2,3) which do not divide 25. The least multiples of 2, 3 not seen already are 22, 27 respectively, so a(21)=22.
%e A357963 a(29), a(30)=42, 28 and every prime dividing 28 (2,7) also divides 42, so a(31) is 56, the least novel multiple of 14 (squarefree kernel of 28).
%t A357963 Block[{a, c, f, g, k, m, q, nn}, nn = 68; c[_] = False; q[_] = 1; Array[Set[{a[#], c[#]}, {#, True}] &, 2]; q[2] = 2; Do[m = FactorInteger[a[n - 1]][[All, 1]]; f = Select[m, CoprimeQ[#, a[n - 2]] &]; If[Length[f] == 0, While[Set[k, # q[#]]; c[k], q[#]++] &[Times @@ m], Set[{k, q[#1]}, {#2, #2/#1}] & @@ First@ MinimalBy[Map[{#, Set[g, q[#]]; While[c[g #], g++]; # g} &, f], Last] ]; Set[{a[n], c[k]}, {k, True}], {n, 3, nn}]; Array[a, nn] ] (* _Michael De Vlieger_, Oct 22 2022 *)
%Y A357963 Cf. A001221, A064413, A352187, A336957, A357942, A357992, A357994, A358267.
%K A357963 nonn
%O A357963 1,2
%A A357963 _David James Sycamore_, Oct 22 2022
%E A357963 More terms from _Michael De Vlieger_, Oct 22 2022