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A357991 Lexicographically earliest counterclockwise square spiral constructed using the nonnegative integers so that a(n) plus all other numbers currently visible from the current number form a distinct sum; start with a(0) = 0.

%I A357991 #10 Oct 24 2022 11:11:58
%S A357991 0,1,1,1,2,1,3,0,4,0,0,0,1,5,0,6,0,0,1,0,2,4,0,7,0,8,0,7,0,7,0,0,0,0,
%T A357991 0,0,0,12,0,13,0,16,1,0,0,0,1,0,0,1,0,0,0,1,0,0,1,12,0,22,0,19,0,20,1,
%U A357991 0,0,0,0,0,0,0,0,17,0,25,0,24,0,20,1,26,0,28,0,26,0,31,0,31,0,0,0,0
%N A357991 Lexicographically earliest counterclockwise square spiral constructed using the nonnegative integers so that a(n) plus all other numbers currently visible from the current number form a distinct sum; start with a(0) = 0.
%C A357991 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 A357991 In the first 50000 terms the smallest number that has not appeared is 9; it is unknown if all the positive numbers eventually appear.
%e A357991 The spiral begins:
%e A357991 .
%e A357991                        .
%e A357991                        .
%e A357991    0---6---0---5---1   7
%e A357991    |               |   |
%e A357991    0   2---1---1   0   0
%e A357991    |   |       |   |   |
%e A357991    1   1   0---1   0   7
%e A357991    |   |           |   |
%e A357991    0   3---0---4---0   0
%e A357991    |                   |
%e A357991    2---4---0---7---0---8
%e A357991 .
%e A357991 .
%e A357991 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 1,0), and 1 (at 0,1). These three numbers sum to 3, so a(6) = 3 so that 3 + 3 = 6, the smallest sum that has not previous occurred.
%e A357991 a(8) = 4 as from square 8, at (1,-1) relative to the starting square, the numbers currently visible are 0 (at 0,-1), 1 (at -1,0), 0 (at 0,0), 1 (at 1,0), and 1 (at 0,1). These five numbers sum to 3, so a(8) = 4 so that 3 + 4 = 7, the smallest sum that has not previous occurred. Note that a(7) = 0 and forms a sum of 8.
%Y A357991 Cf. A357985, A307834, A275609, A274640, A355270.
%K A357991 nonn
%O A357991 0,5
%A A357991 _Scott R. Shannon_, Oct 23 2022