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.

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, 105, 55, 26, 42, 165, 13, 28, 330, 33, 91, 40, 66, 63, 65, 44, 84, 45, 121, 56, 75, 81, 32, 50, 135, 147, 64, 80, 189, 99, 100, 70, 231, 39, 130, 154, 51, 195, 308, 17, 585, 462, 34, 325, 693

Offset: 1

Author

David James Sycamore, Jul 09 2024

Keywords

• nonn

Comments

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).

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.

Examples

			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).
a(4) = 5 because (2,5) = 1 and 2*3*5 = 30 = A002110(3).
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).
		

Links

• Dominic McCarty, Table of n, a(n) for n = 1..10000
• Dominic McCarty, Java program for A374351
• Michael De Vlieger, Log log scatterplot of a(n), 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.
• Michael De Vlieger, Plot prime(i)^m | n at (x,y) = (n,i), 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.

Crossrefs

Cf. A002110, A007947, A055932, A374445.

Programs

Java

See Dominic McCarty link.

Mathematica

nn = 120; c[_] := False;
Array[Set[{a[#], c[#]}, {#, True}] &, 2]; i = a[1]; j = a[2]; u = 3;
f[x_] := f[x] = Or[IntegerQ@ Log2[x], And[EvenQ[x], Union@ Differences@ PrimePi@ FactorInteger[x][[All, 1]] == {1}]];
Do[k = u; While[Or[! CoprimeQ[i, k], c[k], ! f[i*j*k]], k++];
  Set[{a[n], c[k], i, j}, {k, True, j, k}];
  If[k == u, While[c[u], u++]], {n, 3, nn}];
Array[a, nn] (* Michael De Vlieger, Jun 27 2025 *)

Extensions

More terms from Dominic McCarty, Aug 19 2024