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A383342 Lexicographically earliest infinite sequence of distinct positive integers such that the number following any consecutive pair x, y of terms is the smallest novel number divisible by R(x,y) = rad(x*y)/rad(gcd(x,y)).

%I A383342 #26 Jun 21 2025 19:58:32
%S A383342 1,2,4,3,6,8,9,12,10,15,18,20,30,21,70,60,42,35,90,84,105,40,126,210,
%T A383342 5,168,420,25,252,630,45,14,840,75,28,1050,120,7,1260,150,49,1470,180,
%U A383342 56,315,240,98,525,270,112,735,300,140,63,330,770,147,660,1540,189
%N A383342 Lexicographically earliest infinite sequence of distinct positive integers such that the number following any consecutive pair x, y of terms is the smallest novel number divisible by R(x,y) = rad(x*y)/rad(gcd(x,y)).
%C A383342 For numbers x, y let f(x) = {prime p: p|x} and f(y) = {prime q: q|y} be the sets of distinct prime factors of x and y respectively, then R(x,y) is the product of the primes which occur once only in the union of f(x) and f(y). There are two conditions for entry of a prime following adjacent terms x, y: (i). rad(x) = rad(y) implies that the next term is the smallest number not yet seen in the sequence (which could be prime); (ii). rad(x) = A002110(k), rad(y) = A002110(k)/prime(m), m <= k implies prime(m). After computation of 2^24 terms the only primes found are 2,3,5,7. Furthermore in the same data range, if a given composite squarefree number k appears there seems no guarantee that all numbers m with rad(m) = rad(k) will also appear (a(5) = 6 and subsequently we see only 12 and 18; a(9) = 10 and thereafter we see only 20,40,50,80,100). Also the primorial numbers do not appear in order (e.g. A002110(11) appears before A002110(10)).
%H A383342 Michael De Vlieger, <a href="/A383342/b383342.txt">Table of n, a(n) for n = 1..10000</a>
%H A383342 Michael De Vlieger, <a href="/A383342/a383342.png">Log log scatterplot of a(n)</a>, n = 1..2^20.
%H A383342 Michael De Vlieger, <a href="/A383342/a383342_1.png">Log log scatterplot of a(n)</a>, n = 1..2^16, showing primes in red, proper prime powers in gold, squarefree composites in green, and numbers neither squarefree nor prime powers in blue and purple, where purple indicates powerful numbers that are not prime powers. Primorials are highlighted in large bright green points.
%H A383342 Michael De Vlieger, <a href="/A383342/a383342_2.png">Plot p^k | a(n) at (x,y) = (n, pi(p))</a> for n = 1..2^11, 12X vertical exaggeration, with a color function showing k = 1 in black, k = 2 in red, ... maximum value of k in reference range in magenta. The color bar under the plot indicates numbers as immediately above, red = prime, etc.
%e A383342 It follows from the definition that the first two terms must be a(1) = 1, a(2) = 2. R(1,2) = rad(2)/rad(1) = 2 and since 2 is already a term, a(3) = 4.
%e A383342 Since a(2) = 2 and a(3) = 4 have the same rad it follows that a(4) = 3, the smallest novel number.
%e A383342 R(4,3) = rad(12)/rad(1) = 6, so a(5) = 6, since 6 has not occurred earlier.
%e A383342 R(3,6) = rad(18)/rad(3) = 2, so a(6) = 8, the least novel multiple of 2.
%e A383342 R(6,8) = rad(48)/rad(2) = 6/2 = 3 so a(7) = 9, the least novel multiple of 3.
%e A383342 a(23) = 126, a(24) = 210 and R(126,210) = rad(126*210)/rad(42) = 210/42 = 5, which has not occurred earlier, so a(25) = 5.
%t A383342 nn = 120; rad[x_] := rad[x] = Times @@ FactorInteger[x][[All, 1]];
%t A383342 c[_] := False; m[_] := 1; i = 1; j = 2; c[1] = c[2] = True;
%t A383342 {1, 2}~Join~Reap[Do[k = rad[i*j]/rad[GCD[i, j]];
%t A383342   While[c[k*m[k]], m[k]++]; k *= m[k];
%t A383342   Set[{c[k], i, j}, {True, j, k}]; Sow[k],
%t A383342 {n, 3, nn}] ][[-1, 1]] (* _Michael De Vlieger_, Apr 25 2025 *)
%Y A383342 Cf. A002110, A005117, A007947, A362855, A368133, A369825.
%K A383342 nonn
%O A383342 1,2
%A A383342 _David James Sycamore_ and _Michael De Vlieger_, Apr 22 2025