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

Author

Scott R. Shannon, May 28 2022

Keywords

• nonn
• look

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. In the first 200000 terms the smallest unused prime is 181, while the smallest unused composite number is 11881, which is itself a prime power.

Examples

			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.
		

Links

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

Crossrefs

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