A364280 Lexicographically earliest sequence of distinct positive integers such that a(n) is the least novel multiple of m, the product of all primes q < gpf(a(n-2)*a(n-1)) which do not divide a(n-2)*a(n-1); a(1) = 1, a(2) = 2.

1, 2, 3, 4, 5, 6, 7, 10, 9, 8, 11, 105, 12, 13, 385, 18, 14, 15, 16, 17, 15015, 20, 19, 51051, 30, 21, 22, 25, 42, 23, 230945, 84, 24, 35, 26, 33, 70, 27, 28, 40, 36, 29, 37182145, 48, 31, 1078282205, 54, 32, 34, 30030, 37, 6678671, 60060, 38, 51, 5005, 44, 39

Offset: 1

Author

David James Sycamore, Jul 17 2023

Keywords

• nonn

Comments

It follows from the definition that the sequence is infinite.

Let r(n) = a(n-2)*a(n-1).

If rad(r(n)) is a primorial, then every prime q < gpf(r(n)) divides r(n), so m = 1, the empty product, and a(n) = u, the smallest missing number in the sequence so far.

If rad(r(n)) is not a primorial, then m > 1, and significant spikes can occur in scatterplot when there are multiple primes < gpf(r(n)) which do not divide r(n) (e.g., a(12) = 105, a(15) = 385, a(21) = 15015).

The only way a prime can occur is as u.

The sequence is a permutation of the positive integers since no number appears more than once and m = 1 eventually introduces any number not already placed consequent to terms arising from m > 1.

Examples

			a(4) = 4, a(5) = 5, and 3 is the only prime < 5 which does not divide 20, therefore m = 3 and a(6) = 6 since 3 has occurred once already.
a(10) = 8, a(11) = 11 and the product of all primes < 11 which do not divide 8*11 = 88 is 3*5*7 = 105, which has not occurred earlier, so a(12) = 105.
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..6187
• Michael De Vlieger, Log log scatterplot of log_10(a(n)), n = 1..2^16, highlighting prime a(n) in red.

Crossrefs

Cf. A002110, A007947, A359804, A364154.

Programs

Mathematica

nn = 120; c[] := False; m[] := 1; a[1] = i = 1; a[2] = j = 2; c[1] = c[2] = True;
  f[x_] := Times @@ Complement[Prime@ Range[PrimePi@ #[[-1]] - 1], #] &[
    FactorInteger[x][[All, 1]]];
  Do[While[Set[k, f[i j]]; c[k m[k]], m[k]++]; k *= m[k];
    Set[{a[n], c[k], i, j}, {k, True, j, k}], {n, 3, nn}];
Array[a, nn] (* Michael De Vlieger, Jul 17 2023 *)

Extensions

More terms from Michael De Vlieger, Jul 17 2023