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Counterclockwise square spiral constructed using the integers so that a(n) plus all other numbers currently visible from the current number equals n; start with a(0) = 0.
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%I #43 Apr 13 2023 06:08:55

%S 0,1,1,1,2,1,3,-1,6,-2,-1,0,1,9,-8,15,-5,-7,-10,14,-29,58,-78,101,

%T -118,150,-61,309,-307,553,-494,-186,-644,315,-1177,731,-1458,3480,

%U -5183,7096,-8328,9735,-10882,7200,-29452,31322,-52670,51401,-65210,61001,11318,135012,-109687,259226,-221542

%N Counterclockwise square spiral constructed using the integers so that a(n) plus all other numbers currently visible from the current number equals n; start with a(0) = 0.

%C A number is visible from the current number if, given that it has coordinates (x,y) relative to the current number, the greatest common divisor of |x| and |y| is 1.

%C The magnitude of the numbers grow surprisingly quickly, e.g., a(150) = -4346232663618226. The only known terms that equal zero are a(0) and a(11); it is unknown whether more exist or if all integers eventually appear.

%e The spiral begins:

%e .

%e .

%e .

%e -5....15...-8....9.....1 553

%e | | |

%e -7 2....1.....1 0 -307

%e | | | | |

%e -10 1 0.....1 -1 309

%e | | | |

%e 14 3...-1.....6... -2 -61

%e | |

%e -29...58...-78...101...-118...150

%e .

%e .

%e a(6) = 3 as from square 6, at (-1,1) relative to the starting square, the numbers currently visible are 1 (at -1,0), 0 (at 0,0), 1 (at 0,1), and 1 (at 1,0). These four numbers sum to 3, so a(6) = 3 so that 3 + 3 = 6.

%e a(7) = -1 as from square 7, at (0,-1) relative to the starting square, the numbers currently visible are 3 (at -1,-1), 1 (at -1,0), 2 (at -1,1), 0 (at 0,0), 1 (at 1,1), and 1 (at 1,0). These six numbers sum to 8, so a(7) = -1 so that -1 + 8 = 7.

%Y Cf. A357991, A307834, A275609, A274640, A355270.

%K sign

%O 0,5

%A _Scott R. Shannon_, Oct 23 2022