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

1,2

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.

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.

Example

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)

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 *)

Crossrefs

Cf. A002110, A007947, A055932, A362889.

Sequence in context: A185725 A359804 A362889 * A374404 A097347 A356867

Adjacent sequences: A374376 A374377 A374378 * A374380 A374381 A374382

Keyword

nonn

Author

David James Sycamore, Jul 06 2024

Status

approved