A354717 Lexicographically earliest infinite sequence of distinct positive integers such that in any run of four consecutive terms there is one term which is prime to the other three, none of which are pairwise coprime.

1, 2, 4, 6, 5, 8, 12, 14, 11, 10, 16, 18, 7, 20, 15, 24, 13, 3, 9, 21, 17, 27, 30, 33, 19, 22, 36, 26, 23, 28, 32, 34, 25, 38, 42, 44, 29, 40, 46, 48, 31, 50, 45, 35, 37, 55, 60, 65, 41, 39, 52, 54, 43, 56, 58, 62, 47, 64, 66, 68, 49, 51, 72, 57, 53, 63, 69, 75

Offset: 1

Author

David James Sycamore, Jun 03 2022

Keywords

• nonn

Comments

Can be regarded as the reverse of A354732, which has the opposite coprime relations to those defined here. Records tend to be nonprime, but not all nonprimes are records.

The primes do not appear in natural order (5 and 7 precede 3).

Open question: Is it true that in any run of four consecutive terms there is always a prime or prime power (this being the term prime to the other three)?

Conjecture: Sequence is a permutation of the positive integers.

Examples

			a(1,2,3,4) = 1,2,4,6 is the lexicographically earliest string of four consecutive numbers which satisfy the definition, hence the sequence starts with these terms.
a(13,14,15) = 7,20,15 respectively, and 24 is the least unused number such that 7 is prime to 20,15 and 24, whereas (20,15)=5, (15,24)=3 and (20,24)=2. Therefore a(16)=24.
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..10000
• Michael De Vlieger, Scatterplot of a(n), n = 1..512, showing primes in red, odd composites in gold, and evens in blue.

Crossrefs

Cf. A098550, A336957, A352950, A354732.

Programs

Mathematica

Block[{a, c, k, len, u, nn}, nn = 120; c[] = 0; MapIndexed[Set[{a[First[#2]], c[#1]}, {#1, First[#2]}] &, {1, 2, 4, 6}]; len = u = 3; Do[k = u; While[Nand[c[k] == 0, Union@ Tally@ Map[Count[#, 1] &, Outer[GCD, #, #]] == {{1, len}, {len, 1}} &@ {a[i - 3], a[i - 2], a[i - 1], k}], k++]; Set[{a[i], c[k]}, {k, i}], {i, len + 2, nn}]; Array[a, nn] ] (* _Michael De Vlieger, Jun 05 2022 *)