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

Author

David James Sycamore, Jul 08 2024

Keywords

• nonn

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.

Examples

			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).
		

Links

• Rémy Sigrist, Table of n, a(n) for n = 1..10000
• Rémy Sigrist, PARI program
• Michael De Vlieger, Log log scatterplot of a(n), n = 1..10000, showing primes in red, proper prime powers in gold, squarefree composites in green, and numbers neither squarefree nor prime powers in blue and magenta, with magenta representing powerful numbers that are not prime powers.

Crossrefs

Cf. A002110, A007947, A047228, A374351.

Programs

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 *)

PARI

\\ See Links section.

Extensions

More terms from Rémy Sigrist, Jul 11 2024