A257112 Arrange numbers in a clockwise spiral with initial terms a(1)=1, a(2)=2, a(4)=4, a(6)=6, a(8)=8, a(11)=3, a(15)=5, a(19)=7, a(23)=9; thereafter each number is relatively prime to all of its four (N,S,E,W) neighbors, but shares a factor with each of its (N,S,E,W) neighbors at distance 2 and also satisfies an additional condition stated in the comments.

1, 2, 11, 4, 55, 6, 25, 8, 165, 14, 3, 16, 15, 26, 5, 12, 35, 18, 7, 22, 21, 32, 9, 28, 27, 10, 33, 20, 77, 34, 49, 38, 231, 46, 121, 24, 143, 36, 65, 44, 45, 52, 51, 58, 75, 56, 39, 40, 57, 50, 63, 62, 69, 64, 81, 68, 87, 17, 93, 136, 105, 74, 85, 42, 95, 48, 115, 54, 161

Offset: 1

Author

Vladimir Shevelev, Apr 24 2015

Keywords

• nonn

Comments

To formulate the additional condition, let us call two numbers strictly connected if the set of prime divisors of one of them is a subset of the set of prime divisors of the other. Then the positions of two strictly connected terms should not be a knight's move apart.

Start with smallest number which has not yet appeared and satisfies the conditions: a(3)=11; thereafter always choose smallest number which has not yet appeared and satisfies the conditions.

This is a two-dimensional spiral analog of A098550.

In A098550 we have initial terms in the positions 1,2,3.

In the two-dimensional case we have 4 sides. So the initial TERMS are

9

8

7 6 1 2 3 (1)

4

5

But the POSITIONS in the spiral are indexed thus:

.

7--8--9--10

|

6 1--2

| |

5--4--3

.

So the initial terms, by (1), are a(1)=1, a(2)=2, a(4)=4, a(6)=6, a(8)=8, ...

Conjecture: the sequence is a permutation of the positive integers. - Vladimir Shevelev, May 06 2015

Examples

			The spiral begins
.
   21---32----9---28---27---10  etc.
    |
   22   25----8--165---14
    |    |              |
    7    6    1----2    3
    |    |         |    |
   18   55----4---11   16
    |                   |
   35---12----5---26---15
.
Formally the smallest a(12) is 10, but then 10 and 5 are strictly connected numbers on a knight move (and a(13) would not exist). So the smallest suitable a(12)=16.
		

Links

• Peter J. C. Moses, Table of n, a(n) for n = 1..5625
• Peter J. C. Moses, The first few squares.

Crossrefs

Cf. A098550, A257321-A257340, A255370.

Extensions

More terms from Peter J. C. Moses, Apr 29 2015