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.

2, 3, 5, 7, 11, 29, 31, 59, 89, 127, 131, 157, 191, 193, 223, 227, 251, 257, 409, 521, 719, 757, 797, 809, 877, 881, 967, 971, 1009, 1013, 1049, 1087, 1091, 1117, 1123, 1277, 1301, 1361, 1409, 1423, 1447, 1451, 1523, 1531, 1657, 1693, 1697, 1699, 5273, 5323

Offset

1,1

Links

Table of n, a(n) for n=1..50.

Example

Hexagonal grid with integers up to 85:
                         29<---28<---27<---26<-7,25<=6,24<==5/23
                        /                       /              \\
                      30                      8                4/22
                     /                       /                    \\
                  31,53<-52<---51<---50<--9,49<--48<---47         3,21
                  /  \                    /              \        /  \
                54    32                10               1,46--->2    20
               /        \              /                    \           \
           55,79<--78<-33,77<--76<-11,75<--74<---73          45          19
            //             \           \           \           \        /
        56,80               34          12          72          44    18
         //                   \           \           \        /  \  /
     57,81                     35          13--->14->15,71-->16-->17,43
      //                         \                    /           /
  58,82                           36                70          42
   //                               \              /           /
59,83                                37--->38->39,69-->40--->41
   \\                                           /
  60,84                                       68
     \\                                      /
    61,85--->62--->63--->64--->65--->66--->67
Prime number (p), square of the distance (s) from p to origin, and index (n) in the sequence for p up to 71 are:
p:  2  3  5  7  11  13  17  19  23  29  31  37  41  43  47  53   59   61  67  71
s:  1  3  7  9  13  13   9   7   7  37  43  31  19   9   1  43  109  109  43   7
n:  1  2  3  4   5  --  --  --  --   6   7  --  --  --  --  --    8   --  --  --

Prog

(Python)
from sympy import isprime
dx = [2, 1, -1, -2, -1, 1]; dy = [0, 1, 1, 0, -1, -1]
x = 0; y = 0; rec = 0; d = 0
for n in range(2, 10001):
    if isprime(n-1) == 1: d += 1; d %= 6
    x += dx[d]; y += dy[d]; s = x*x + 3*y*y
    if isprime(n) == 1 and s > rec: print(n); rec = s

Crossrefs

Cf. A113519, A257745, A309755, A335916.

Sequence in context: A167119 A165682 A296924 * A175711 A099160 A181503

Adjacent sequences: A343420 A343421 A343422 * A343424 A343425 A343426

Keyword

nonn

Author

Ya-Ping Lu, Apr 15 2021

Status

approved