A382951 Sequence of positive integers with no repetitions and, when put in a spiral, all lines (straight or diagonal) are pairwise coprime.

1, 2, 3, 5, 4, 7, 11, 9, 13, 17, 19, 23, 8, 29, 31, 27, 25, 37, 39, 16, 41, 43, 14, 47, 33, 53, 35, 59, 61, 67, 71, 73, 49, 79, 83, 89, 97, 101, 103, 55, 107, 109, 91, 113, 85, 127, 131, 137, 139, 121, 149, 151, 157, 133, 163, 65, 167, 51, 125, 173, 143, 179, 181, 191, 161, 22, 193, 169, 197, 199, 211

Offset: 1

Author

Bryle Morga, Apr 09 2025

Keywords

• nonn

Comments

We take the lexicographically earliest sequence that fits the name.

It seems likely (but unproven) that every positive integers appear and that this sequence is a permutation of the positive integers. Some number just takes very long to appear. For instance, here are the number of steps it took to reach some numbers:

6: 160 steps

10: 468 steps

12: 571 steps

15: 4048 steps!!

18: 582 steps

20: 1492 steps

21: 820 steps

It takes at least floor(N/2)^2 steps before the first N integers appear. Any tighter bound?

Examples

			    4 -- 5 -- 3
    |         |
    7    1 -- 2
    |
   11 --
.
Look at the 7th term. It couldn't be 1, 2, 3, 4, 5, and 7 as they already occurred in the sequence. It also can't be 6, 8, or 10 because they all share factors with 4 which is in the vertical line (...4, 7, 11...). It cannot be a 9 because of the diagonal (...11, 1, 3...).
		

Crossrefs

Cf. A336349.

Programs

Mathematica

S[n_]:=Block[{v,sq={1}, p=Most[{Re@#, Im@#}&/@ Fold[Join[#1, Last[#1]+I^#2 Range[#2/2]]&, {0}, Range[4n+2]]], A=<||>, T=<||>, s, d=Rest@ Tuples[{0,1,-1}, 2]}, T[1]=1; A[{0,0}]=1; s[z_]:=Block[{L={},o}, Do[o=z; While[ Max[Abs[o+e]]<=n, AppendTo[L,o+=e]],{e,d}];L]; Do[v=LCM@@ A/@ Intersection[Keys[A], s[u]]; k=2; While[ KeyExistsQ[T,k] || GCD[v,k]>1,k++]; AppendTo[sq,k]; T[k]=1; A[u]=k, {u, Rest@p}]; (* Print@ Graphics@ Table[ Text[sq[[i]], p[[i]]], {i,Length[p]}]; *) sq]; S[4] (* S[n] returns the values for a grid of semidiameter n. Uncomment the Print to show the spiral. Giovanni Resta, Apr 10 2025 *)