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A271725 T(n,k) is an array read by rows, with n > 0 and k=1..4, where row n gives four prime numbers in increasing order with locations in right angles of each concentric square drawn on a distorted version of the Ulam spiral.

%I A271725 #14 Jun 16 2017 02:54:09
%S A271725 3,7,17,19,13,23,37,41,307,359,401,419,13807,14159,14401,14519,41413,
%T A271725 42023,42437,42641,6317683,6325223,6330257,6332771,22958473,22972847,
%U A271725 22982437,22987229,39081253,39100007,39112517,39118769,110617807,110649359,110670401,110680919
%N A271725 T(n,k) is an array read by rows, with n > 0 and k=1..4, where row n gives four prime numbers in increasing order with locations in right angles of each concentric square drawn on a distorted version of the Ulam spiral.
%C A271725 See the illustration for more information.
%C A271725 Conjecture: there is an infinity of concentric squares having a prime number in each right angle. The number 5 is the center of all the squares.
%C A271725 It seems that the drawing of an infinite number of concentric squares having a prime number in each corner is impossible in an Ulam spiral. But with a slight distortion of this space, the problem becomes possible.
%C A271725 The illustration (see the link) shows the new version of a spiral with two remarkable orthogonal diagonals containing four classes of prime numbers given by the sequences A125202, A121326, A028871 and A073337 supported by four line segments. These intersect at a single point represented by the prime number 5.
%C A271725 The sequence of the corresponding length of the sides is {s(k)} = {2, 4, 18, 118, 204, 2514, 4792, 6252, 10518, 14032, 16752, 17598, ...}
%C A271725 The primes are defined by the polynomials: [4*m^2-10*m+7, (2*m-1)^2-2, 4*m^2+1, 4*(m+1)^2-6*(m+1)+1]. The sequence of the corresponding m is {b(k)} = {2, 3, 10, 60, 103, 1258, 2397, 3127, 5260, 7017, 8377, 8800, 10375, 11518, 11523, 12498, 15415, 15888, ...} with the relation b(k) = 1 + s(k)/2.
%C A271725 The array begins:
%C A271725       3,     7,    17,    19;
%C A271725      13,    23,    37,    41;
%C A271725     307,   359,   401,   419;
%C A271725   13807, 14159, 14401, 14519;
%C A271725   41413, 42023, 42437, 42641;
%C A271725   ...
%C A271725 Construction of the spiral (see the illustration in the link):
%C A271725                 .  .  .  .  .  .  .  .  .  .  .  .
%C A271725                .  42  41  40  39  38  37   .  .  .
%C A271725                                        |
%C A271725                .  43  20  19  18  17  36  35  .  .
%C A271725                                    |
%C A271725                .  .   21   6   5  16  15  34  .  .
%C A271725                                |
%C A271725                .  .   22   7   4   3  14  33  .  .
%C A271725                .  .   23   8   1   2  13  32  .  .
%C A271725                .  .   24   9  10  11  12  31  .  .
%C A271725                .  .   25  26  27  28  29  30  .  .
%C A271725                    .  .  .  .  .  .  .  .  .  .  .
%C A271725 The first squares of center 5 having a prime number in each vertex are:
%C A271725     19  18  17      41  40  39  38  37
%C A271725      6   5  16      20  19  18  17  36
%C A271725      7   4  3       21   6   5  16  15   . . . .
%C A271725                     22   7   4   3  14
%C A271725                     23   8   1   2  13
%H A271725 Michel Lagneau, <a href="/A271725/a271725.pdf">Illustration</a>
%p A271725 for n from 1 to 10000 do :
%p A271725   x1:=4*n^2-10*n+7:x2:=(2*n-1)^2-2:
%p A271725   x3:=4*(n+1)^2-6*(n+1)+1:x4:=4*n^2+1:
%p A271725    if isprime(x1) and isprime(x2) and isprime(x3) and isprime(x4)
%p A271725     then
%p A271725      printf("%d %d %d %d %d \n",n,x1,x2,x4,x3):
%p A271725     else
%p A271725     fi:
%p A271725 od:
%Y A271725 Cf. A028871, A073337, A121326, A125202, A200975.
%K A271725 nonn
%O A271725 1,1
%A A271725 _Michel Lagneau_, Apr 13 2016