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Lexicographically earliest sequence of distinct positive integers on a square spiral such that any 3X3 square of numbers sums to a prime.
8

%I #13 May 30 2022 08:23:32

%S 1,2,3,4,5,6,7,8,11,9,10,13,12,14,20,16,15,17,19,22,18,21,25,26,35,23,

%T 24,27,28,30,29,31,33,37,41,36,32,34,43,38,40,52,39,42,66,48,45,44,46,

%U 47,49,54,50,56,51,57,53,55,61,72,67,59,58,62,60,63,71,68,74,76,70,80,64,65,69,77,73

%N Lexicographically earliest sequence of distinct positive integers on a square spiral such that any 3X3 square of numbers sums to a prime.

%C See A354442 for the successive prime sums formed by each completed 3X3 square of numbers.

%H Scott R. Shannon, <a href="/A354441/a354441.png">Image of the first 4000 terms</a>. The green line is y = n.

%e The spiral begins

%e .

%e .

%e 32--36--41--37--33--31--29 57

%e | | |

%e 34 15--16--20--14--12 30 51

%e | | | | |

%e 43 17 5---4---3 13 28 56

%e | | | | | | |

%e 38 19 6 1---2 10 27 50

%e | | | | | |

%e 40 22 7---8--11---9 24 54

%e | | | |

%e 52 18--21--25--26--35--23 49

%e | |

%e 39--42--66--48--45--44--46--47

%e .

%e .

%e a(9) = 11 as this completes a 3X3 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.

%e a(12) = 13 as this completes a 3X3 square of numbers 8,11,9,1,2,10,4,3,13, which sum to 61, a prime, and 13 is the smallest unused number to form a prime sum.

%Y Cf. A354442, A337116, A257339, A354434, A000040.

%K nonn

%O 1,2

%A _Scott R. Shannon_, May 29 2022