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A362631 Lexicographically earliest infinite sequence of distinct positive integers with a(n) = n for n <= 3, and for n > 3 a(n) is the least novel multiple of the greatest prime divisor of a(n-2) which does not divide a(n-1).

%I A362631 #26 Jun 23 2023 07:51:00
%S A362631 1,2,3,4,6,5,9,10,12,15,8,20,7,25,14,30,21,35,18,28,24,42,11,49,22,56,
%T A362631 33,63,44,70,55,77,40,66,45,88,50,99,60,110,27,121,36,132,13,143,16,
%U A362631 26,17,39,34,52,51,65,68,78,85,91,102,104,119,117,136,130,153,156,170,169,187,182,204,195,221,75,238,80,255,32,272,19
%N A362631 Lexicographically earliest infinite sequence of distinct positive integers with a(n) = n for n <= 3, and for n > 3  a(n) is the least novel multiple of the greatest prime divisor of a(n-2) which does not divide a(n-1).
%C A362631 The definition reflects that of A098550 in that it places a condition on a(n-2) which does not apply to a(n-1).
%C A362631 If there is no prime divisor of a(n-2) which does not divide a(n-1), then by empty product convention a(n) = u the least unused number.
%C A362631 Some primes (23,29,31,47,...) enter because of dividing a(n-2) but not a(n-1), whereas others (5,7,11,13,17,19,...) enter as least u; see Example.
%C A362631 With the exception of 16 all least u terms (up to a(2^28)) are primes, so it seems likely that a(47) = 16 is a one-off (fluke) term.
%C A362631 The scatterplot resembles a fine-toothed comb similar to those of A361629, A361133 and A361534, in which each "tooth" starts with a novel prime p and continues with a run of (mostly) alternate multiples of p and the greatest prime less than p until interrupted by the arrival of the next prime, and so forth.
%C A362631 The sequence, after a(1) = 1 can be represented as an irregular table in which the n-th row starts with prime(n), see Example.
%C A362631 Conjectured to be a permutation of the positive integers with the primes in order.
%H A362631 Michael De Vlieger, <a href="/A362631/b362631.txt">Table of n, a(n) for n = 1..10000</a>
%H A362631 Michael De Vlieger, <a href="/A362631/a362631.png">Scatterplot of a(n)</a> n = 1..10000.
%H A362631 Michael De Vlieger, <a href="/A362631/a362631_1.png">Log log scatterplot of a(n)</a> n = 1..2^20.
%H A362631 Michael De Vlieger, <a href="/A362631/a362631_2.png">Log log scatterplot of a(n)</a>, n = 1..10000, showing primes in red, composite prime powers in gold, squarefree composites in green, and numbers neither squarefree nor prime powers in blue. We accentuate powerful numbers that are not prime powers in light blue.
%e A362631 a(4,5) = 4, 6 and since rad(4)|rad(6) a(6) = least u = 5.
%e A362631 a(11,12) = 8, 20 and since rad(8)|rad(20) a(13) = least u = 7.
%e A362631 a(44,45) = 132, 13 and gpd(132) = 11 does not divide 13, and since it is the 13th occurrence of p = 11, a(46) = 13*11 = 143.
%e A362631 a(45,46) = 13, 143 which forces a(47) = least u = 16 (see Comment).
%e A362631 a(90,91) = 69, 114 and 23 is the greatest prime dividing 69 which does not divide 114. Since 23 has not appeared earlier in the sequence a(92) = 23.
%e A362631 There is as yet no known formula for the row lengths of the table below. Whereas most rows terminate with a multiple of the prime they start with, there are exceptions, e.g., 47, 109. This behavior is open to explanation.
%e A362631 The table starts:
%e A362631 2;
%e A362631 3,4,6;
%e A362631 5,9,10,12,15,8,20;
%e A362631 7,25,14,30,21,35,18,28,24,42;
%e A362631 11,49,22,56,33,63,44,70,55,77,40,66,45,88,50,99,60,110,27,121,36,132;
%e A362631 13,143,16,26;
%e A362631 17,39,34,52,51,65,68,78,85,91,102,104,119,117,136,130,153,156,170,169,187,182,204,195,221,75,238,80,255,32,272;
%e A362631 19,....
%t A362631 mm = 3; nn = 10^4; c[_] := False; q[_] := 1;
%t A362631 Array[Set[{a[#], c[#]}, {#, True}] &, mm]; q[2]++; u = mm + 1;
%t A362631 i = a[mm - 1]; j = a[mm]; s = {1};
%t A362631 Do[t = FactorInteger[i][[All, 1]];
%t A362631   If[SubsetQ[s, t], q[u]++; Set[k, u],
%t A362631    p = SelectFirst[Reverse@ t, CoprimeQ[j, #] &];
%t A362631    If[! IntegerQ[p], Set[k, u], While[c[p*q[p]], q[p]++]; k = p*q[p]] ];
%t A362631   Set[{a[n], c[k], i, j, s}, {k, True, j, k, t}];
%t A362631   If[k == u, While[c[u], u++]], {n, mm + 1, nn}];
%t A362631 Array[a, nn]
%Y A362631 Cf. A007947, A098550, A361629, A361133, A361534, A362855.
%K A362631 nonn
%O A362631 1,2
%A A362631 _David James Sycamore_, May 06 2023