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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. 1
3, 7, 17, 19, 13, 23, 37, 41, 307, 359, 401, 419, 13807, 14159, 14401, 14519, 41413, 42023, 42437, 42641, 6317683, 6325223, 6330257, 6332771, 22958473, 22972847, 22982437, 22987229, 39081253, 39100007, 39112517, 39118769, 110617807, 110649359, 110670401, 110680919 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

See the illustration for more information.

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.

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.

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.

The sequence of the corresponding length of the sides is {s(k)} = {2, 4, 18, 118, 204, 2514, 4792, 6252, 10518, 14032, 16752, 17598, ...}

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.

The array begins:

      3,     7,    17,    19;

     13,    23,    37,    41;

    307,   359,   401,   419;

  13807, 14159, 14401, 14519;

  41413, 42023, 42437, 42641;

  ...

Construction of the spiral (see the illustration in the link):

                .  .  .  .  .  .  .  .  .  .  .  .

               .  42  41  40  39  38  37   .  .  .

                                       |

               .  43  20  19  18  17  36  35  .  .

                                   |

               .  .   21   6   5  16  15  34  .  .

                               |

               .  .   22   7   4   3  14  33  .  .

               .  .   23   8   1   2  13  32  .  .

               .  .   24   9  10  11  12  31  .  .

               .  .   25  26  27  28  29  30  .  .

                   .  .  .  .  .  .  .  .  .  .  .

The first squares of center 5 having a prime number in each vertex are:

    19  18  17      41  40  39  38  37

     6   5  16      20  19  18  17  36

     7   4  3       21   6   5  16  15   . . . .

                    22   7   4   3  14

                    23   8   1   2  13

LINKS

Table of n, a(n) for n=1..36.

Michel Lagneau, Illustration

MAPLE

for n from 1 to 10000 do :

  x1:=4*n^2-10*n+7:x2:=(2*n-1)^2-2:

  x3:=4*(n+1)^2-6*(n+1)+1:x4:=4*n^2+1:

   if isprime(x1) and isprime(x2) and isprime(x3) and isprime(x4)

    then

     printf("%d %d %d %d %d \n", n, x1, x2, x4, x3):

    else

    fi:

od:

CROSSREFS

Cf. A028871, A073337, A121326, A125202, A200975.

Sequence in context: A305348 A191147 A227211 * A058887 A306355 A087749

Adjacent sequences:  A271722 A271723 A271724 * A271726 A271727 A271728

KEYWORD

nonn

AUTHOR

Michel Lagneau, Apr 13 2016

STATUS

approved

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Last modified May 31 22:45 EDT 2020. Contains 334756 sequences. (Running on oeis4.)