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%I #7 Oct 23 2022 23:37:28
%S 1,2,4,6,3,9,12,8,10,5,15,18,14,7,21,24,16,20,25,30,27,33,11,22,26,13,
%T 39,36,28,35,40,32,34,17,51,42,49,56,38,19,57,45,50,44,55,60,48,54,66,
%U 77,63,69,23,46,52,65,70,84,72,78,91,98,58,29,87,75,80,62
%N 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 greatest least multiple of any such p which has not already occurred. Otherwise a(n) is the least novel multiple of the squarefree kernel of a(n-1). (see comments).
%C 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) = Max{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 has not already occurred.
%C Companion to A357963, (which uses "Min" rather than "Max" in selection of a(n)). The first departure from A357963 occurs at a(21) because a(19),a(20) = 25,30, and 30 has two divisors (2,3) which do not divide 25. Of these the least multiples not occurring already are 22, and 27 respectively. At this point in A357963 22 is the chosen term, whereas here a(21) = 27. This has the effect of temporarily reversing (for the next prime = 11) the normal way primes seem to arrive in this sequence (2p,p,3p, as in A064413). Thus we see 30,27,33,11,22 (3p,p,2p). This may happen elsewhere in the data, consequent to choice of "Max" over "Min".
%C Conjecture: Permutation of the positive integers; primes being in natural order, and the slowest numbers to appear (as in A352187).
%H Michael De Vlieger, <a href="/A357994/a357994.png">Scatterplot of a(n)</a>, n = 1..120, showing primes p in red, 2p in blue, and 3p in green, for comparison to A064413.
%H Michael De Vlieger, <a href="/A357994/a357994_1.png">Scatterplot of a(n)</a>, n = 1..2^20.
%H Michael De Vlieger, <a href="/A357994/a357994_2.png">Log-log scatterplot of a(n)</a>, n = 1..2^14, labeling records in red and local minima in blue, highlighting primes in green, prime squares in light blue, and other prime powers in gold. Demonstrates delayed 17^2 and 19^2, early records such as a(268) = 455.
%e a(1),a(2)=1,2 and 2 is the only prime dividing 2 which does not divide 1, therefore a(3)=4, the least multiple of 2 which has not occurred already. In this case (as in all terms up to and including a(20), "Max" gives the same term as "Min".
%t 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@ MaximalBy[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 23 2022 *)
%Y Cf. A001221, A064413, A352187, A357963.
%K nonn
%O 1,2
%A _David James Sycamore_, Oct 23 2022
%E More terms by _Michael De Vlieger_, Oct 23 2022
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