A354434 a(1) = 1; for n > 1, a(n) is the smallest unused square spiral number such that a(n) shares a factor with all existing numbers in its Moore neighborhood.

1, 2, 4, 6, 3, 9, 12, 18, 8, 10, 14, 16, 20, 22, 24, 15, 21, 27, 30, 33, 36, 39, 26, 28, 32, 34, 38, 40, 42, 44, 46, 48, 50, 54, 45, 35, 7, 63, 51, 57, 60, 66, 52, 72, 78, 84, 56, 58, 62, 64, 68, 70, 74, 76, 80, 82, 86, 88, 90, 75, 96, 100, 105, 49, 77, 91, 119, 102, 69, 81, 108, 92, 94, 98, 104

Offset

1,2

Comments

The sequence is conjectured to be a permutation of the positive integers, although it takes many terms for most primes to appear, e.g. a(1807) = 13, a(35156) = 179. The primes do not occur in their natural order.

After a million terms the numbers 397, 419, 443, 463, 467, 487, 499, 503, 509, ... are still missing - Scott R. Shannon, Jun 20 2026.

Links

Scott R. Shannon, Table of n, a(n) for n = 1..20000

Michael S. Branicky, Python program for OEIS A354434 and A397168

Scott R. Shannon, Table of n, a(n) for n = 1..200000

Scott R. Shannon, Table of n, a(n) for n = 1..10^6

Scott R. Shannon, Image of the first 200000 terms. The green line is y = n.

Scott R. Shannon, Image of the first 500000 terms on square the spiral. The number colors are graduated across the spectrum, from red to violet, to show their relative size.

Example

The spiral begins
                                .
                                .
    7--35--45--54--50--48--46  82
    |                       |   |
   63  21--15--24--22--20  44  80
    |   |               |   |   |
   51  27   3---6---4  16  42  76
    |   |   |       |   |   |   |
   57  30   9   1---2  14  40  74
    |   |   |           |   |   |
   60  33  12--18---8--10  38  70
    |   |                   |   |
   66  36--39--26--28--32--34  68
    |                           |
   52--72--78--84--56--58--62--64
.
.
a(11) = 14 as the existing numbers in the Moore neighborhood when a(11) is being placed are 4,2,8,10, and 14 is the smallest unused number that shares a factor with all these numbers.

Crossrefs

Cf. A336946, A064413, A253279, A257112, A257339.

See A397168 for the inverse sequence.

Sequence in context: A291577 A393113 A395077 * A362842 A338059 A397460

Adjacent sequences: A354431 A354432 A354433 * A354435 A354436 A354437

Keyword

nonn,look,nice,changed

Author

Scott R. Shannon, May 28 2022

Status

approved