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
Keywords
Examples
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 -- -- --
Programs
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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