%I #19 Jun 19 2021 15:03:32
%S 2,3,5,7,11,29,31,59,89,127,131,157,191,193,223,227,251,257,409,521,
%T 719,757,797,809,877,881,967,971,1009,1013,1049,1087,1091,1117,1123,
%U 1277,1301,1361,1409,1423,1447,1451,1523,1531,1657,1693,1697,1699,5273,5323
%N 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 Hexagonal grid with integers up to 85:
%e 29<---28<---27<---26<-7,25<=6,24<==5/23
%e / / \\
%e 30 8 4/22
%e / / \\
%e 31,53<-52<---51<---50<--9,49<--48<---47 3,21
%e / \ / \ / \
%e 54 32 10 1,46--->2 20
%e / \ / \ \
%e 55,79<--78<-33,77<--76<-11,75<--74<---73 45 19
%e // \ \ \ \ /
%e 56,80 34 12 72 44 18
%e // \ \ \ / \ /
%e 57,81 35 13--->14->15,71-->16-->17,43
%e // \ / /
%e 58,82 36 70 42
%e // \ / /
%e 59,83 37--->38->39,69-->40--->41
%e \\ /
%e 60,84 68
%e \\ /
%e 61,85--->62--->63--->64--->65--->66--->67
%e 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 p: 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71
%e s: 1 3 7 9 13 13 9 7 7 37 43 31 19 9 1 43 109 109 43 7
%e n: 1 2 3 4 5 -- -- -- -- 6 7 -- -- -- -- -- 8 -- -- --
%o (Python)
%o from sympy import isprime
%o dx = [2, 1, -1, -2, -1, 1]; dy = [0, 1, 1, 0, -1, -1]
%o x = 0; y = 0; rec = 0; d = 0
%o for n in range(2, 10001):
%o if isprime(n-1) == 1: d += 1; d %= 6
%o x += dx[d]; y += dy[d]; s = x*x + 3*y*y
%o if isprime(n) == 1 and s > rec: print(n); rec = s
%Y Cf. A113519, A257745, A309755, A335916.
%K nonn
%O 1,1
%A _Ya-Ping Lu_, Apr 15 2021
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