

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.


6



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
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OFFSET

1,2


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 twodimensional spiral analog of A098550.
In A098550 we have initial terms in the positions 1,2,3.
In the twodimensional 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:
.
78910

6 12
 
543
.
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


LINKS

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


EXAMPLE

The spiral begins
.
21329282710 etc.

22 25816514
  
7 6 12 3
   
18 55411 16
 
351252615
.
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.


CROSSREFS

Cf. A098550, A257321A257340, A255370.
Sequence in context: A070532 A038217 A152985 * A077272 A009301 A087552
Adjacent sequences: A257109 A257110 A257111 * A257113 A257114 A257115


KEYWORD

nonn


AUTHOR

Vladimir Shevelev, Apr 24 2015


EXTENSIONS

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


STATUS

approved



