A364246 a(1) = 1. Thereafter, a(n) is the least novel multiple of either prime(k+1) if rad(a(n-1)) = A002110(k), or Product_{prime q; q < gpf(a(n-1)); and q!|a(n-1)} q otherwise.

1, 2, 3, 4, 6, 5, 12, 10, 9, 8, 15, 14, 30, 7, 60, 21, 20, 18, 25, 24, 35, 36, 40, 27, 16, 33, 70, 39, 770, 42, 45, 22, 105, 26, 1155, 28, 75, 32, 48, 50, 51, 10010, 54, 55, 84, 65, 462, 80, 57, 170170, 63, 90, 49, 120, 56, 135, 34, 15015, 38, 255255, 44, 210

Offset: 1

Author

David James Sycamore, Jul 15 2023

Keywords

• nonn

Comments

In other words, if the squarefree kernel of a(n-1) is a primorial term then a(n) is the least novel multiple of the smallest prime which does not divide a(n-1). Otherwise a(n) is the least novel multiple of the product of all primes < gpd(a(n-1)) which do not divide a(n-1). Primes >= 5 arrive late (as least unused term), and a(k) is prime(m) iff a(k-1) is A002110(m-1). The pattern around a prime is P(k), prime(k+1), 2*P(k), m*prime(k+1) for some multiplier m, where P(k) = A002110(k). The sequence is conjectured to be a permutation of the positive integers, with primes in natural order.

A common mode in this sequence is alternation of squarefree semiprime q(j)*q(k), j < k, followed by P(k-1)/q(j). The alternation often occurs in runs such that each iteration increments k. Example: a(241..246): q(2)*q(17) -> P(16)/q(2) -> q(2)*q(18) -> P(17)/q(2) -> q(2)*q(19) -> P(18)/q(2). a(16539..16572) represents a run of 17 alternations. - Michael De Vlieger, Jul 17 2023

Examples

			a(5) = 6 a primorial number so the next term is the smallest prime not dividing 6, thus a(7) = 5.
a(26) = 33 = 3*11 and the product of primes < 11 which do not divide 11 is 2*5*7 = 70, which has not occurred previously, therefore a(27) = 70.
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..10000
• Michael De Vlieger, Log log scatterplot of log_10(a(n)), n = 1..2^16, highlighting prime a(n) in red.

Crossrefs

Cf. A002110, A006530, A007947, A362889, A364154.

Programs

Mathematica

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

Extensions

More terms from Michael De Vlieger, Jul 17 2023