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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.

%I A374445 #28 Jun 24 2025 12:17:13
%S A374445 1,2,3,4,9,8,15,14,45,16,27,10,21,20,63,40,81,32,75,28,135,56,165,98,
%T A374445 225,64,105,22,315,44,525,88,735,128,243,50,147,80,189,100,231,130,
%U A374445 693,160,441,110,273,220,567,200,729,70,33,140,99,280,297,350,363
%N 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.
%C A374445 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.
%C A374445 {a(n); n >= 2} is conjectured to be a permutation of A047228.
%H A374445 Rémy Sigrist, <a href="/A374445/b374445.txt">Table of n, a(n) for n = 1..10000</a>
%H A374445 Rémy Sigrist, <a href="/A374445/a374445.gp.txt">PARI program</a>
%H A374445 Michael De Vlieger, <a href="/A374445/a374445.png">Log log scatterplot of a(n)</a>, 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.
%e A374445 The sequence starts with a(1) = 1, a(2) = 2 since (1,2) = 1 and 1*2 = A002110(1).
%e A374445 a(3) = 3 since (2,3) = 1 and 2*3 = 6 = A002110(2).
%t A374445 nn = 540; c[_] := False;
%t A374445 Array[Set[{a[#], c[#]}, {#, True}] &, 2]; j = a[2]; u = 3;
%t A374445 f[x_] := f[x] = Or[IntegerQ@ Log2[x], And[EvenQ[x], Union@ Differences@ PrimePi@ FactorInteger[x][[All, 1]] == {1}]];
%t A374445 Monitor[Do[k = u;
%t A374445   While[Or[! CoprimeQ[j, k], c[k], ! f[j*k]], k++];
%t A374445   Set[{a[n], c[k], j}, {k, True, k}];
%t A374445   If[k == u, While[c[u], u++]], {n, 3, nn}], n];
%t A374445 Array[a, nn] (* _Michael De Vlieger_, Jul 16 2024 *)
%o A374445 (PARI) \\ See Links section.
%Y A374445 Cf. A002110, A007947, A047228, A374351.
%K A374445 nonn
%O A374445 1,2
%A A374445 _David James Sycamore_, Jul 08 2024
%E A374445 More terms from _Rémy Sigrist_, Jul 11 2024