A374379 a(1) = 1, a(2) = 2. Let i = a(n-2), j = a(n-1), then for n > 2 a(n) is the least novel k such that rad(k) !| rad(i*j) whereas rad(i*j*k) is a primorial number (A002110), where rad is A007947.

1, 2, 3, 5, 4, 6, 10, 7, 9, 20, 14, 12, 15, 21, 8, 25, 18, 28, 30, 11, 35, 24, 22, 70, 27, 33, 140, 13, 66, 105, 26, 44, 210, 39, 55, 42, 52, 110, 63, 65, 88, 84, 40, 77, 36, 45, 49, 16, 60, 56, 99, 50, 91, 132, 75, 98, 121, 90, 112, 143, 120, 119, 286, 135, 126, 154, 80, 48, 147

Offset: 1

Author

David James Sycamore, Jul 06 2024

Keywords

• nonn

Comments

For n > 2 a(n) is prime iff rad(i*j) is a primorial number, for then k is the smallest prime which does not divide rad(i*j).

Initially same terms as A362889, but divergence occurs at a(69) = 147.

Conjectured to be a permutation of the natural numbers with primes in order.

Examples

			a(3) = 3 since rad(3) !| rad(1*2) whereas rad (1*2*3) = 6 = A002110(2).
a(5,6) = {4,6}—> a(7) = 10 since rad(10) !| rad(24), rad(4*6*10) = 30 = A002110(3) and there is no smaller novel term with this property.
From _Michael De Vlieger_, Jul 06 2024: (Start)
Table of a(3..18) showing prime decomposition of rad(i*j) and rad(i*j*k):
   n  a(n)   rad(i*j)   rad(i*j*k)
  -------------------------------
   3    3    2          2 3
   4    5    2 3        2 3 5
   5    4    . 3 5      2 3 5
   6    6    2 . 5      2 3 5
   7   10    2 3        2 3 5
   8    7    2 3 5      2 3 5 7
   9    9    2 . 5 7    2 3 5 7
  10   20    . 3 . 7    2 3 5 7
  11   14    2 3 5      2 3 5 7
  12   12    2 . 5 7    2 3 5 7
  13   15    2 3 . 7    2 3 5 7
  14   21    2 3 5      2 3 5 7
  15    8    . 3 5 7    2 3 5 7
  16   25    2 3 . 7    2 3 5 7
  17   18    2 . 5      2 3 5
  18   28    2 3 5      2 3 5 7   (End)
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..2050
• Michael De Vlieger, Log log scatterplot of log_10(a(n)), n = 1..2048.

Crossrefs

Cf. A002110, A007947, A055932, A362889.

Programs

Mathematica

nn = 1200; c[] := False; rad[n] := rad[n] = Times @@ FactorInteger[n][[All, 1]];
Array[Set[{a[#], c[#]}, {#, True}] &, 2];
f[n_] := Or[IntegerQ@ Log2[n],
  And[EvenQ[n], Union@ Differences@ Map[PrimePi, FactorInteger[n][[All, 1]]] == {1}]]; i = a[1]; j = a[2]; u = 3;
Monitor[Do[r = rad[i*j]; k = u;
  While[Or[c[k], Divisible[#, rad[k]], ! f[#  k]] &[i*j], k++];
  Set[{a[n], c[k], i, j}, {k, True, j, k}];
  If[k == u, While[c[u], u++]], {n, 3, nn}], n];
Array[a, nn] (* Michael De Vlieger, Jul 06 2024 *)