login
The lowest number on Ulam Spiral for which all numbers in the square which is centered at a(n) and spans n-1 spaces in each cardinal direction are nonprime.
2

%I #37 Nov 06 2022 09:05:24

%S 1,26,1016,5136,39639,203100,2729736,32264250,42119062,1065799391,

%T 12444190246

%N The lowest number on Ulam Spiral for which all numbers in the square which is centered at a(n) and spans n-1 spaces in each cardinal direction are nonprime.

%C The n-th term creates a (2*n-1) X (2*n-1) square centered at a(n).

%H Samuel Harkness, <a href="/A357376/a357376.jpg">Illustration of terms 1 through 4</a>

%H Rémy Sigrist, <a href="/A357376/a357376.txt">C++ program</a>

%e For n=2, test squares with 1 layer of numbers around the center number until a square is enclosed with only nonprimes.

%e .

%e 65 64 63 62 61 60 59 58 57

%e .

%e 66 37 36 35 34 33 32 31 56

%e .

%e 67 38 17 16 15 14 13 30 55

%e .

%e 68 39 18 5 4 3 12 29 54

%e .

%e 69 40 19 6 1 2 11 28 53

%e . +--------------+

%e 70 41 20 7 8 9 | 10 27 52 |

%e . | |

%e 71 42 21 22 23 24 | 25 26 51 |

%e . | |

%e 72 43 44 45 46 47 | 48 49 50 |

%e . +--------------+

%e 73 74 75 76 77 78 79 80 81

%e .

%e The first number tested, 1, does not work because 2, 3, 5 and 7 are in the 1-layered square and are prime. The first square that works is 26 because 10, 27, 52, 25, 26, 51, 48, 49, and 50 are all nonprime.

%o (C++) See Links section.

%Y Cf. A000040, A115258, A136626, A225073.

%K nonn,more

%O 1,2

%A _Samuel Harkness_, Sep 26 2022

%E a(8)-a(11) from _Rémy Sigrist_, Sep 27 2022