A374445 - OEIS

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

(list; graph; refs; listen; history; text; internal format)

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

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.

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 *)