%I #13 Aug 10 2020 01:26:30

%S 1,2,4,6,3,9,12,8,10,5,20,14,7,28,16,18,15,21,24,22,11,33,30,25,35,40,

%T 45,36,26,42,27,63,48,32,34,50,38,54,39,51,60,44,46,66,55,65,70,49,56,

%U 52,58,72,57,76,62,78,13,117,69,75,80,64,68,74,37,148

%N a(1) = 1; for n > 1, a(n) is the next square spiral number not already used such that a(n) shares a factor with a(n-1) and also with the adjacent number on the inner spiral arm if such a number exists.

%C This is a variation of the EKG sequence A064413 where the numbers are written on the square spiral such that each new number must share a common factor with not only the previous number but also with the adjacent inner spiral number, in one of the four axial directions, if such a number exists. This additional restriction causes the numbers to violate some of the patterns the numbers form in the standard EKG sequence, e.g., an odd prime p does not need to be preceded by 2p or followed by 3p, and the primes do not appear in increasing order.

%C For the first 100000 terms the smallest unseen number is 433.

%H Scott R. Shannon, <a href="/A336946/a336946.png">Line graph of the first 1000 terms</a>.

%e The spiral begins

%e .

%e .

%e 38--50--34--32--48--63--27 78

%e | | |

%e 54 15--18--16--28---7 42 62

%e | | | | |

%e 39 21 3---6---4 14 26 76

%e | | | | | | |

%e 51 24 9 1---2 20 36 57

%e | | | | | |

%e 60 22 12---8--10---5 45 72

%e | | | |

%e 44 11--33--30--25--35--40 58

%e | |

%e 46--66--55--65--70--49--56--52

%e .

%e a(1)-a(8) = 1,2,4,6,3,9,12,8. The adjacent inner spiral number is 1 which all numbers share a factor with so the numbers are the same as A064413(n).

%e a(9) = 10. This is the first number that must have a common factor with two numbers, the previous number a(8) = 8 and the adjacent spiral number a(2) = 2. The lowest unused number satisfying this requirement is 10.

%e a(10) = 5. As this number is on the corner of a square spiral arm it only needs to share a divisor with a(9) = 10. The lowest unseen number satisfying this is 5.

%e a(11) = 20. This number must have a common factor with the previous number a(10) = 5 and the adjacent spiral number a(2) = 2. The lowest unused number satisfying this requirement is 20. This is also the first number to differ from A064413 which only needs to find the lowest unused number sharing a factor with 5, which is 15.

%Y Cf. A064413, A073734, A253279, A257112.

%K nonn

%O 1,2

%A _Scott R. Shannon_, Aug 08 2020