A336349 -id:A336349 - cp's OEIS frontend

A336350 Square spiral of distinct nonnegative integers constructed by greedy algorithm, such that two terms on the same row or on the same column have no common one bit in their binary representations.

0, 1, 2, 4, 8, 6, 16, 3, 12, 32, 24, 64, 5, 48, 72, 128, 256, 17, 96, 512, 10, 33, 144, 320, 1024, 516, 192, 1280, 2048, 4096, 9, 130, 1536, 288, 2112, 4100, 8192, 514, 160, 6144, 16384, 32768, 13, 1152, 768, 10240, 20480, 65536, 18, 32800, 12288, 2304, 640

Offset: 0

Rémy Sigrist, Jul 19 2020

We can always extend the sequence with a power of 2 greater than any previous term, so the sequence is well defined.

For symmetry reasons, we obtain the same sequence when considering a clockwise or a counterclockwise square spiral, or when initially moving towards any unit direction.

			The spiral begins:
         264------80--262144---81920----5120---32896----2560----8224-------7
           |                                                               |
       49152    8192----4100----2112-----288----1536-----130-------9   65552
           |       |                                               |       |
        3072     514     256-----128------72------48-------5    4096  131072
           |       |       |                               |       |       |
        4608     160      17       8-------4-------2      64    2048   17408
           |       |       |       |               |       |       |       |
       73728    6144      96       6       0-------1      24    1280     640
           |       |       |       |                       |       |       |
      393216   16384     512      16-------3------12------32     192    2304
           |       |       |                                       |       |
      524288   32768      10------33-----144-----320----1024-----516   12288
           |       |                                                       |
     1048576      13----1152-----768---10240---20480---65536------18---32800
           |
          19---65792---18432----9216---33280--655360--266240-2097152-1048584

Rémy Sigrist, Table of n, a(n) for n = 0..10200
Rémy Sigrist, Colored representation of the spiral for -250 <= x <= 250 and -250 <= y <= 250 (where the hue is function of a(n) and black pixels indicate powers of 2)
Rémy Sigrist, PARI program for A336350

See A336349 for a similar sequence.

PARI
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A366303 Square array A(n, k), n, k > 0, read and filled by upwards antidiagonals the greedy way with distinct positive integers such that any two distinct terms in the same row or column are coprime.

Original entry on oeis.org

1, 2, 3, 5, 7, 4, 9, 8, 11, 13, 17, 19, 21, 15, 23, 29, 25, 31, 37, 41, 35, 43, 47, 53, 14, 59, 61, 67, 49, 71, 65, 73, 55, 79, 83, 89, 97, 101, 103, 107, 6, 109, 113, 127, 121, 131, 137, 139, 149, 119, 143, 151, 157, 163, 167, 169, 173, 179, 181, 191, 12, 133, 193, 197, 199, 211

Offset: 1

Rémy Sigrist, Oct 06 2023

This sequence is a variant of A366030, with one less constraint.
All the prime numbers appear in the sequence, in ascending order.
For any prime number p, the first multiple of p in the sequence is p.
Will every positive integer appear in the sequence?

			Array A(n, k) begins:
  n\k |   1    2    3    4    5    6    7    8    9   10
  ----+-------------------------------------------------
    1 |   1    3    4   13   23   35   67   89  121  167
    2 |   2    7   11   15   41   61   83  127  163  199
    3 |   5    8   21   37   59   79  113  157  197  247
    4 |   9   19   31   14   55  109  151  193  221  307
    5 |  17   25   53   73    6  143  133  241  293  353
    6 |  29   47   65  107  119   12  239  283  349  401
    7 |  43   71  103  149  191  233   10   33  161  463
    8 |  49  101  139  181  229  281   27   16   95  451
    9 |  97  137  179  227  277  323  341   91   18  115
   10 | 131  173  223  253  347  397  461   85  613   24

Rémy Sigrist, Colored representation of the array for n, k <= 881 (grayish pixels correspond to prime numbers)
Rémy Sigrist, PARI program

Cf. A336349, A366030, A366304.

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

Original entry on oeis.org

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

Bryle Morga, Apr 09 2025

nonn

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:
 160 steps
 468 steps
 571 steps
4048 steps!!
 582 steps
1492 steps
 820 steps
It takes at least floor(N/2)^2 steps before the first N integers appear. Any tighter bound?

			    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...).

Cf. A336349.

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 *)

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A336350 Square spiral of distinct nonnegative integers constructed by greedy algorithm, such that two terms on the same row or on the same column have no common one bit in their binary representations.

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A366303 Square array A(n, k), n, k > 0, read and filled by upwards antidiagonals the greedy way with distinct positive integers such that any two distinct terms in the same row or column are coprime.

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PARI

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

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Mathematica