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A374351 Lexicographically earliest infinite sequence of distinct positive integers such that for any triple i,j,k of consecutive terms, gcd(i,k) = 1 and A007947(i*j*k) is a term in A002110.

%I A374351 #51 Jun 27 2025 19:43:38
%S A374351 1,2,3,5,4,6,9,25,8,12,15,7,14,30,11,49,60,10,77,21,20,16,27,35,22,18,
%T A374351 105,55,26,42,165,13,28,330,33,91,40,66,63,65,44,84,45,121,56,75,81,
%U A374351 32,50,135,147,64,80,189,99,100,70,231,39,130,154,51,195,308,17,585,462,34,325,693
%N A374351 Lexicographically earliest infinite sequence of distinct positive integers such that for any triple i,j,k of consecutive terms, gcd(i,k) = 1 and A007947(i*j*k) is a term in A002110.
%C A374351 In other words a(n) is least k such that (k, a(n-2)) = 1 and rad(a(n-2)*a(n-1)*k) is a primorial number (alternatively i*j*k is in A055932).
%C A374351 Conjectures: A permutation of the positive integers, a(n) = prime p (> 2) iff p is least unused odd term, 2|i, and rad(i*j) is a primorial number divisible by all primes < p but not by p; primes appear in natural order.
%H A374351 Dominic McCarty, <a href="/A374351/b374351.txt">Table of n, a(n) for n = 1..10000</a>
%H A374351 Dominic McCarty, <a href="/A374351/a374351_1.txt">Java program for A374351</a>
%H A374351 Michael De Vlieger, <a href="/A374351/a374351.png">Log log scatterplot of a(n)</a>, n = 1..10^4, showing primes in red, perfect prime powers in gold, squarefree composites in green, and numbers neither squarefree nor prime powers in blue and purple, where purple represents powerful numbers that are not prime powers.
%H A374351 Michael De Vlieger, <a href="/A374351/a374351_1.png">Plot prime(i)^m | n at (x,y) = (n,i)</a>, n = 1..2048, 8X vertical exaggeration, with a color function showing m = 1 in black, m = 2 in red, m = 3 in orange, ..., largest m in the dataset in magenta.
%e A374351 a(1,2,3) = 1,2,3 the lexicographically earliest triple of numbers satisfying the definition: 3 and 1 are coprime whilst 1*2*3 = 6 = A002110(2).
%e A374351 a(4) = 5 because (2,5) = 1 and 2*3*5 = 30 = A002110(3).
%e A374351 a(9,10,11) = 8,12,15 and 7 is the smallest novel number prime to 12 and rad(12*15*7) = 210 = A002110(4).
%t A374351 nn = 120; c[_] := False;
%t A374351 Array[Set[{a[#], c[#]}, {#, True}] &, 2]; i = a[1]; j = a[2]; u = 3;
%t A374351 f[x_] := f[x] = Or[IntegerQ@ Log2[x], And[EvenQ[x], Union@ Differences@ PrimePi@ FactorInteger[x][[All, 1]] == {1}]];
%t A374351 Do[k = u; While[Or[! CoprimeQ[i, k], c[k], ! f[i*j*k]], k++];
%t A374351   Set[{a[n], c[k], i, j}, {k, True, j, k}];
%t A374351   If[k == u, While[c[u], u++]], {n, 3, nn}];
%t A374351 Array[a, nn] (* _Michael De Vlieger_, Jun 27 2025 *)
%o A374351 (Java) See Dominic McCarty link.
%Y A374351 Cf. A002110, A007947, A055932, A374445.
%K A374351 nonn
%O A374351 1,2
%A A374351 _David James Sycamore_, Jul 09 2024
%E A374351 More terms from _Dominic McCarty_, Aug 19 2024