A309755 Primes with record Euclidean distance from the origin. When starting rightwards in a grid, turn left after a prime number, if not walk straight on.

2, 3, 11, 29, 59, 97, 149, 151, 191, 193, 211, 223, 239, 263, 281, 307, 311, 331, 337, 593, 613, 631, 641, 653, 659, 853, 857, 877, 881, 907, 911, 967, 971, 991, 997, 1801, 1811, 1847, 1861, 1901, 1907, 2251, 2267, 2281, 2287, 2309, 2311, 2657, 2671, 2677, 3163, 3167, 3187, 3191, 3299, 3319, 3343, 3691, 3697, 3719, 3727

Offset

1,1

Comments

Cells can contain more than one number.

This sequence differs from A309701, where the Manhattan distance is taken.

Links

Pieter Post, Table of n, a(n) for n = 1..182

Ya-Ping Lu, Positions of integers up to 1 million and terms less than 100 million

Example

Grid of the first 34 steps. 0 (second cell in sixth row) 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 two steps away from the origin, 3 (2,1) is at a distance of sqrt(5). Next record distance is 11 (4,-1), at distance sqrt(17). Next is 29 (4,5), at distance sqrt(41).

Mathematica

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 @ (r^2); If[r1 > rm, rm = r1; AppendTo[s, p]]; q = p, {n, 0, 3000}]; s (* Amiram Eldar, Aug 15 2019 *)

Prog

(Python)
from sympy import isprime
n, g, h=[], [1, 0, -1, 0], [0, 1, 0, -1]
z=10000
ca, cb, cc=2, 0, 0
for j in range(2, z):
    if isprime(j):
        cc=cc+1
        cd, ce=g[cc%4], h[cc%4]
    ca, cb=ca+cd, cb+ce
    n.append([j+1, ca, cb, ca**2+cb**2])
print(2, end=', ')
v=4
for j in n:
    if j[3]>v and isprime(j[0]):
        print(j[0], end=', ')
        v=j[3]
(PARI) z=0; d=1; m=0; for (n=1, 3727, z+=d; if (isprime(n), d*=I; if (m<norm(z), m=norm(z); print1 (n ", ")))) \\ Rémy Sigrist, Aug 15 2019
(Python)
from sympy import isprime; from math import sqrt; x = y = d = rec = 0
for n in range(1, 3750):
    dx = 1 if d == 0 else -1 if d == 2 else 0; x += dx
    dy = 1 if d == 1 else -1 if d == 3 else 0; y += dy
    if isprime(n):
        d = (d + 1)%4; z = sqrt(x*x + y*y)
        if z > rec: print(n, end = ', '); rec = z # Ya-Ping Lu, Nov 05 2025

Crossrefs

Cf. A000040, A309701.

Sequence in context: A176316 A181956 A237038 * A309701 A243896 A202211

Adjacent sequences: A309752 A309753 A309754 * A309756 A309757 A309758

Keyword

nonn

Author

Pieter Post, Aug 15 2019

Status

approved