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A367082 a(1), a(2) = 2. Thereafter a(n) is the least novel multiple of the greatest prime which divides precisely one of a(n-1), a(n-2) but not the other. If no such prime exists a(n) is the least novel multiple of the smallest prime dividing neither a(n-1) nor a(n-2).

%I A367082 #12 Nov 12 2023 22:08:49
%S A367082 1,2,4,3,6,8,9,12,10,5,14,7,16,21,28,15,35,42,20,49,56,18,63,70,25,77,
%T A367082 11,84,22,33,24,44,55,30,66,88,27,99,110,40,121,132,36,143,13,154,26,
%U A367082 39,45,52,65,50,78,91,98,104,117,48,130,156,60,169,182,105,195,208,75,221,17,234,34,51,54,68,85,80,102,119
%N A367082 a(1), a(2) = 2. Thereafter a(n) is the least novel multiple of the greatest prime which divides precisely one of a(n-1), a(n-2) but not the other. If no such prime exists a(n) is the least novel multiple of the smallest prime dividing neither a(n-1) nor a(n-2).
%C A367082 The second condition of the definition applies iff rad(a(n-1)) = rad(a(n-2)). This occurs when n = 3 (a(2) = 2, a(3) = 4), and seems never to occur again.
%C A367082 Conjecture: This is a permutation of the positive integers, with primes in order. Each prime p generates a trajectory T(p), dominated by multiples of p, see graph. Typically T(prime(k)) is initiated following the earliest term divisible by prime(k). Some trajectories are more prominent in the graph than others e.g. T(7),T(11),T(13) extend from p = prime(k) to prime(k+1)*prime(k). Others, e.g. T(3) terminate earlier but include prime(k)^2, whilst some do not reach as far as prime(k)^2 (T(5) has just two terms). Thus three categories of prime trajectory can be distinguished in the graph (respectively "full", "medium" and "small"). The graph resembles an inclined comb whose teeth correspond to full and medium trajectories; see Example.
%H A367082 Michael De Vlieger, <a href="/A367082/b367082.txt">Table of n, a(n) for n = 1..10000</a>
%H A367082 Michael De Vlieger, <a href="/A367082/a367082.png">Log log scatterplot of a(n)</a>, n = 1..2^20.
%H A367082 Michael De Vlieger, <a href="/A367082/a367082_1.png">Log log scatterplot of a(n)</a>, n = 1..2^12, showing primes in red, composite prime powers in gold, squarefree composites in green, and numbers neither squarefree nor prime powers in blue.
%F A367082 If T(prime(k)) is a fully developed trajectory is contains (prime(k+1) - 1)*3/2 terms.
%e A367082 a(1,2) = 1,2 so a(3) = 4, the least novel multiple of 2 (which divides 2 but not 1). Since rad(2) = rad(4) = 2 there is no prime which divides one of a(2), a(3) but not the other so by the second condition of the definition a(4) = 3, the least novel multiple of the smallest prime (3) which divides neither a(2) = 2 nor a(3) = 4.
%e A367082 The sequence can be presented as an irregular table where row(k) starts with A008578(k), and with the exception of rows 1 and 2, ends with the earliest multiple of A008578(k+1).
%e A367082 The table starts:
%e A367082   1;
%e A367082   2,4;
%e A367082   3,6,8,9,12,10;
%e A367082   5,14;
%e A367082   7,16,21,28,15,35,42,20,49,56,18,63,70,25,77;
%e A367082   11,84,22,33,24,44,55,30,66,88,27,110,40,121,132,36,143;
%e A367082   13,154.....
%e A367082 T(3) is a medium trajectory, includes 3^2 but not 3*5 = 15, which appears later, in T(7). T(5) is a short trajectory, stopped by 14, does not include 25 which is delayed until T(7); T(7) is the first full trajectory, including 49, and ending with 7*11 = 77. In full and medium trajectories T(p) we see pairs of consecutive multiples of p separated by a multiple of a smaller prime. If T(prime(k)) is full it contains (prime(k+1) - 1) multiples of prime(k) and (prime(k+1) - 1)/2 multiples of smaller primes, thus T(7) contains 15 terms; see Formula.
%t A367082 nn = 120;
%t A367082   c[_] := False; m[_] := 1;
%t A367082   f[x_] := f[x] = FactorInteger[x][[All, 1]];
%t A367082   g[x_] :=
%t A367082    Block[{q = 2},
%t A367082     If[OddQ[x], q, While[Divisible[x, q], q = NextPrime[q] ]; q] ];
%t A367082   Array[Set[{a[#], c[#]}, {#, True}] &, 2];
%t A367082   i = a[1]; j = a[2]; u = 3; ri = {}; rj = {j};
%t A367082   Do[Set[r, Times @@ Union[ri, rj]]; If[ri == rj,
%t A367082     While[c[Set[k, # m[#] ] ], m[#]++] &[g[r]],
%t A367082     While[c[Set[k, # m[#] ] ], m[#]++] &[FactorInteger[r][[-1, 1]] ] ];
%t A367082    Set[{a[n], c[k], i, j, ri, rj}, {k, True, j, k, rj, f[k]}], {n, 3, nn}];
%t A367082 Array[a, nn] (* _Michael De Vlieger_, Nov 06 2023 *)
%Y A367082 Cf. A007947, A008578, A351495, A359804.
%K A367082 nonn
%O A367082 1,2
%A A367082 _David James Sycamore_ and _Michael De Vlieger_, Nov 06 2023