A343423 - cp's OEIS frontend

A343423 Prime numbers p such that Euclidean distance from origin to p in hexagonal grid sets a new record. Number '1' is placed at the origin and '2' at (1, 0). Number 'm' (m >= 3) is placed by moving one unit forward in the direction from 'm-2' to 'm-1', if m - 1 is not a prime; otherwise, making 1/6 turn counterclockwise at 'm-1' followed by moving one unit forward.

%I A343423 #19 Jun 19 2021 15:03:32
%S A343423 2,3,5,7,11,29,31,59,89,127,131,157,191,193,223,227,251,257,409,521,
%T A343423 719,757,797,809,877,881,967,971,1009,1013,1049,1087,1091,1117,1123,
%U A343423 1277,1301,1361,1409,1423,1447,1451,1523,1531,1657,1693,1697,1699,5273,5323
%N A343423 Prime numbers p such that Euclidean distance from origin to p in hexagonal grid sets a new record. Number '1' is placed at the origin and '2' at (1, 0). Number 'm' (m >= 3) is placed by moving one unit forward in the direction from 'm-2' to 'm-1', if m - 1 is not a prime; otherwise, making 1/6 turn counterclockwise at 'm-1' followed by moving one unit forward.
%e A343423 Hexagonal grid with integers up to 85:
%e A343423                          29<---28<---27<---26<-7,25<=6,24<==5/23
%e A343423                         /                       /              \\
%e A343423                       30                      8                4/22
%e A343423                      /                       /                    \\
%e A343423                   31,53<-52<---51<---50<--9,49<--48<---47         3,21
%e A343423                   /  \                    /              \        /  \
%e A343423                 54    32                10               1,46--->2    20
%e A343423                /        \              /                    \           \
%e A343423            55,79<--78<-33,77<--76<-11,75<--74<---73          45          19
%e A343423             //             \           \           \           \        /
%e A343423         56,80               34          12          72          44    18
%e A343423          //                   \           \           \        /  \  /
%e A343423      57,81                     35          13--->14->15,71-->16-->17,43
%e A343423       //                         \                    /           /
%e A343423   58,82                           36                70          42
%e A343423    //                               \              /           /
%e A343423 59,83                                37--->38->39,69-->40--->41
%e A343423    \\                                           /
%e A343423   60,84                                       68
%e A343423      \\                                      /
%e A343423     61,85--->62--->63--->64--->65--->66--->67
%e A343423 Prime number (p), square of the distance (s) from p to origin, and index (n) in the sequence for p up to 71 are:
%e A343423 p:  2  3  5  7  11  13  17  19  23  29  31  37  41  43  47  53   59   61  67  71
%e A343423 s:  1  3  7  9  13  13   9   7   7  37  43  31  19   9   1  43  109  109  43   7
%e A343423 n:  1  2  3  4   5  --  --  --  --   6   7  --  --  --  --  --    8   --  --  --
%o A343423 (Python)
%o A343423 from sympy import isprime
%o A343423 dx = [2, 1, -1, -2, -1, 1]; dy = [0, 1, 1, 0, -1, -1]
%o A343423 x = 0; y = 0; rec = 0; d = 0
%o A343423 for n in range(2, 10001):
%o A343423     if isprime(n-1) == 1: d += 1; d %= 6
%o A343423     x += dx[d]; y += dy[d]; s = x*x + 3*y*y
%o A343423     if isprime(n) == 1 and s > rec: print(n); rec = s
%Y A343423 Cf. A113519, A257745, A309755, A335916.
%K A343423 nonn
%O A343423 1,1
%A A343423 _Ya-Ping Lu_, Apr 15 2021

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