A309755 -id:A309755 - cp's OEIS frontend

A309701 Primes with record Manhattan distance from origin. When starting rightwards in a grid, turn left after a prime number. If not, walk straight on.

2, 3, 11, 29, 59, 97, 151, 193, 211, 223, 239, 281, 307, 311, 331, 337, 479, 541, 593, 613, 631, 641, 659, 877, 881, 907, 911, 997, 1409, 1861, 1907, 2267, 2281, 2287, 2309, 2311, 2503, 2579, 2609, 2617, 2657, 2671, 2677, 3671, 3691, 3697, 3727, 3761, 3767, 3793, 3797, 4201, 4327, 4357, 4391, 4397, 4507, 4721, 4751, 4909

Offset: 1

Pieter Post, Aug 13 2019

nonn

This sequence differs from A309755 where the Euclidean distance is used.

			Grid of the first 34 steps. 0 represents (0,0).
xx  xx   xx   31   30    29
xx  xx   xx   32   xx    28
xx  xx   xx   33   xx    27
xx  xx   xx   34   xx    26
xx 5/17 4/16 3/15  14   13/25
x 0/6/18  1    2   xx   12/24
xx 7/19 8/20 9/21 10/22 11/23
2 (2,0) is 2 steps away from the origin, 3 (2,1) has a distance of 3. Next record distance is 11 (4,-1), distance 5. Next is 29 (4,5), distance 9.

Pieter Post, Table of n, a(n) for n = 1..166

Cf. A000040, A309755.

step[n_] := Switch[n, 0, {1,0}, 1, {0,1}, 2, {-1,0}, 3, {0,-1}]; r = {0,0}; q = 0; s={}; rm=0; Do[p = NextPrime[q]; r += step[Mod[n, 4]] * (p-q); r1 = Total @ Abs @ r; If[r1 > rm, rm = r1; AppendTo[s, p]]; q = p, {n, 0, 3000}]; s (* Amiram Eldar, Aug 15 2019 *)

upto(n) = {my(pos = [0, 0], rotateLeft = [0, -1; 1, 0], step = [1, 0], recordDistance = 0, q = 0, res = List(), i = 0); forprime(p = 2, n, pos += (p - q) * step; step *= rotateLeft; if(abs(pos[1]) + abs(pos[2]) > recordDistance, i++; recordDistance = abs(pos[1]) + abs(pos[2]); listput(res, p)); q = p); res} \\ David A. Corneth, Aug 15 2019

def prime(z):
    isPrime=True
    for y in range(2,int(z**0.5)+1) :
        if z%y==0:
            isPrime=False
            break
    return isPrime
m,n,g,h=[],[],[1,0,-1,0],[0,1,0,-1]
for c in range (2,10000):
    if prime(c)==True:
        m.append(c)
ca,cb,cc=2,0,0
for j in range(2,10000):
    if j in m:
        cc=cc+1
        cd,ce=g[cc%4],h[cc%4]
    ca,cb=ca+cd,cb+ce
    n.append([j+1,ca,cb,abs(ca)+abs(cb)])
v=2
for j in n:
    if j[3]>v and j[0] in m:
        print (j)
        v=j[3]

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.

Original entry on oeis.org

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

Ya-Ping Lu, Apr 15 2021

nonn

			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   --  --  --

Cf. A113519, A257745, A309755, A335916.

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

cp's OEIS Frontend

A309701 Primes with record Manhattan distance from origin. When starting rightwards in a grid, turn left after a prime number. If not, walk straight on.

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