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)).

1, 2, 4, 3, 6, 8, 9, 12, 10, 15, 18, 20, 30, 21, 70, 60, 42, 35, 90, 84, 105, 40, 126, 210, 5, 168, 420, 25, 252, 630, 45, 14, 840, 75, 28, 1050, 120, 7, 1260, 150, 49, 1470, 180, 56, 315, 240, 98, 525, 270, 112, 735, 300, 140, 63, 330, 770, 147, 660, 1540, 189

Offset: 1

Author

David James Sycamore and Michael De Vlieger, Apr 22 2025

Keywords

• nonn

Comments

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)).

Examples

			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.
Since a(2) = 2 and a(3) = 4 have the same rad it follows that a(4) = 3, the smallest novel number.
R(4,3) = rad(12)/rad(1) = 6, so a(5) = 6, since 6 has not occurred earlier.
R(3,6) = rad(18)/rad(3) = 2, so a(6) = 8, the least novel multiple of 2.
R(6,8) = rad(48)/rad(2) = 6/2 = 3 so a(7) = 9, the least novel multiple of 3.
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.
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..10000
• Michael De Vlieger, Log log scatterplot of a(n), n = 1..2^20.
• Michael De Vlieger, Log log scatterplot of a(n), 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.
• Michael De Vlieger, Plot p^k | a(n) at (x,y) = (n, pi(p)) 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.

Crossrefs

Cf. A002110, A005117, A007947, A362855, A368133, A369825.

Programs

Mathematica

nn = 120; rad[x_] := rad[x] = Times @@ FactorInteger[x][[All, 1]];
c[] := False; m[] := 1; i = 1; j = 2; c[1] = c[2] = True;
{1, 2}~Join~Reap[Do[k = rad[i*j]/rad[GCD[i, j]];
  While[c[k*m[k]], m[k]++]; k *= m[k];
  Set[{c[k], i, j}, {True, j, k}]; Sow[k],
{n, 3, nn}] ][[-1, 1]] (* Michael De Vlieger, Apr 25 2025 *)