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A354435 Lexicographically earliest sequence of distinct positive integers on a square spiral such that any 3 X 3 square of numbers sums to a prime, and these primes are distinct.

%I A354435 #13 Jun 01 2022 09:59:05
%S A354435 1,2,3,4,5,6,7,8,11,9,10,13,12,14,20,16,15,17,19,22,18,21,25,26,39,23,
%T A354435 24,29,36,30,27,28,34,35,48,31,32,33,42,40,41,37,38,43,44,45,54,46,49,
%U A354435 47,50,60,63,67,53,51,52,55,59,72,75,65,68,81,56,57,58,74,85,61,86,73,62,64,66,90,87
%N A354435 Lexicographically earliest sequence of distinct positive integers on a square spiral such that any 3 X 3 square of numbers sums to a prime, and these primes are distinct.
%C A354435 This sequence uses the same rules as A354453 but here the sum is over every 3 X 3 square of numbers. The terms are widely spread out as in A354453 but here they display an unusual concentration in density along at least three bands that wander between the upper and lower bounds of the terms. See the linked images. The reason for this behavior is unknown.
%C A354435 See A354461 for the successive prime sums formed by each completed 3 X 3 square of numbers.
%H A354435 Scott R. Shannon, <a href="/A354435/a354435.png">Image of the first 200000 terms</a>. The green line is y = n.
%H A354435 Scott R. Shannon, <a href="/A354435/a354435_1.png">Image of the first 1000000 terms</a>.
%e A354435 The spiral begins
%e A354435                                 .
%e A354435                                 .
%e A354435    32--31--48--35--34--28--27  51
%e A354435     |                       |   |
%e A354435    33  15--16--20--14--12  30  53
%e A354435     |   |               |   |   |
%e A354435    42  17   5---4---3  13  36  67
%e A354435     |   |   |       |   |   |   |
%e A354435    40  19   6   1---2  10  29  63
%e A354435     |   |   |           |   |   |
%e A354435    41  22   7---8--11---9  24  60
%e A354435     |   |                   |   |
%e A354435    37  18--21--25--26--39--23  50
%e A354435     |                           |
%e A354435    38--43--44--45--54--46--49--47
%e A354435 .
%e A354435 .
%e A354435 a(9) = 11 as this completes a 3 X 3 square of numbers 5,4,3,6,1,2,7,8,11 which sum to 47, a prime, and 11 is the smallest unused number to form a prime sum that has not occurred before.
%e A354435 a(25) = 39 as this completes a 3 X 3 square of numbers 1,2,10,8,11,9,25,26,39 which sum to 131, a prime, and 39 is the smallest unused number to form a prime sum that has not occurred before. Note that 35 would generate a square sum of 127, also a prime, but 127 was formed previously by the 3 X 3 square 19,6,1,22,7,8,18,21,25 so cannot be used. This is the first term to differ from A354441.
%Y A354435 Cf. A354461, A354441, A354442, A354453, A337116, A000040.
%K A354435 nonn,look
%O A354435 1,2
%A A354435 _Scott R. Shannon_, May 28 2022