A374445 Lexicographically earliest sequence of distinct positive integers such that any pair of consecutive terms are coprime whereas the squarefree kernel of their product is primorial.

1, 2, 3, 4, 9, 8, 15, 14, 45, 16, 27, 10, 21, 20, 63, 40, 81, 32, 75, 28, 135, 56, 165, 98, 225, 64, 105, 22, 315, 44, 525, 88, 735, 128, 243, 50, 147, 80, 189, 100, 231, 130, 693, 160, 441, 110, 273, 220, 567, 200, 729, 70, 33, 140, 99, 280, 297, 350, 363

Offset

1,2

Comments

In other words rad(a(n-2)*a(n-1)) is a term in A002110 whereas a(n-2) and a(n-1) share no common divisor > 1. Every term > a(1) = 1 is divisible by 2 or by 3 but not by both, and all terms other than 1,2,3 are composite.

{a(n); n >= 2} is conjectured to be a permutation of A047228.

Links

Rémy Sigrist, Table of n, a(n) for n = 1..10000

Rémy Sigrist, PARI program

Example

The sequence starts with a(1) = 1, a(2) = 2 since (1,2) = 1 and 1*2 = A002110(1).
a(3) = 3 since (2,3) = 1 and 2*3 = 6 = A002110(2).

Mathematica

nn = 540; c[_] := False;
Array[Set[{a[#], c[#]}, {#, True}] &, 2]; j = a[2]; u = 3;
f[x_] := f[x] = Or[IntegerQ@ Log2[x], And[EvenQ[x], Union@ Differences@ PrimePi@ FactorInteger[x][[All, 1]] == {1}]];
Monitor[Do[k = u;
  While[Or[! CoprimeQ[j, k], c[k], ! f[j*k]], k++];
  Set[{a[n], c[k], j}, {k, True, k}];
  If[k == u, While[c[u], u++]], {n, 3, nn}], n];
Array[a, nn] (* Michael De Vlieger, Jul 16 2024 *)

Prog

(PARI) \\ See Links section.

Crossrefs

Cf. A002110, A007947, A047228, A374351.

Sequence in context: A340807 A329449 A365117 * A227928 A098293 A095260

Adjacent sequences: A374442 A374443 A374444 * A374446 A374447 A374448

Keyword

nonn

Author

David James Sycamore, Jul 08 2024

Extensions

More terms from Rémy Sigrist, Jul 11 2024

Status

approved