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A354453 Lexicographically earliest sequence of distinct positive integers on a square spiral such that any 2 X 2 square of numbers sums to a prime, and that prime is unique for all such squares. Start with a(1) = 0.

%I A354453 #13 May 31 2022 11:38:39
%S A354453 0,1,2,4,3,6,5,8,14,7,9,17,10,12,19,21,11,18,16,32,13,23,25,20,30,15,
%T A354453 27,40,31,43,22,28,39,37,36,41,24,51,57,48,35,69,26,49,66,53,65,58,76,
%U A354453 29,61,88,38,90,33,113,34,54,123,67,86,74,100,98,42,75,91,70,96,102,71,117,44,106,126
%N A354453 Lexicographically earliest sequence of distinct positive integers on a square spiral such that any 2 X 2 square of numbers sums to a prime, and that prime is unique for all such squares. Start with a(1) = 0.
%C A354453 This is a variation of A337116 where the same rules apply except that the primes generated by all 2 X 2 square sums must be unique. This leads to the terms having a far greater variation in value while being concentrated along a central line which shows wave-like variations in density. See the linked image. The reason for this behavior is unknown.
%C A354453 See A354460 for the successive prime sums formed by each completed 2 X 2 square of numbers.
%H A354453 Scott R. Shannon, <a href="/A354453/a354453.png">Image of the first 200000 terms</a>. The green line is y = n.
%e A354453 The spiral begins
%e A354453                                 .
%e A354453                                 .
%e A354453    24--41--36--37--39--28--22 113
%e A354453     |                       |   |
%e A354453    51  11--21--19--12--10  43  33
%e A354453     |   |               |   |   |
%e A354453    57  18   3---4---2  17  31  90
%e A354453     |   |   |       |   |   |   |
%e A354453    48  16   6   0---1   9  40  38
%e A354453     |   |   |           |   |   |
%e A354453    35  32   5---8--14---7  27  88
%e A354453     |   |                   |   |
%e A354453    69  13--23--25--20--30--15  61
%e A354453     |                           |
%e A354453    26--49--66--53--65--58--76--29
%e A354453 .
%e A354453 .
%e A354453 a(9) = 14 as this completes a 2 X 2 square of numbers 0,1,8,14 which sum to 23, a prime, and 14 is the smallest unused number to form a prime sum that has not occurred before. Note that 10 is unused and would form a prime sum of 19, see A337116, but 19 was formed previously by the square 6,0,5,8, so cannot be used. This is the first term to differ from A337116.
%Y A354453 Cf. A354460, A337116, A354441, A257339, A354434, A000040.
%K A354453 nonn,look
%O A354453 1,3
%A A354453 _Scott R. Shannon_, May 30 2022