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 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 (list; graph; refs; listen; history; text; internal format)
 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 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 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 .    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. CROSSREFS Cf. A098550, A257321-A257340, 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

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Last modified May 26 18:08 EDT 2020. Contains 334630 sequences. (Running on oeis4.)