A345293 a(n) is the first number on the n-th layer in a layered square spiral of primes.

2, 73, 149, 211, 307, 467, 659, 839, 1061, 1319, 1511, 1697, 1949, 2129, 2381, 2677, 2819, 3137, 3307, 3407, 3559, 3907, 4079, 4253, 4591, 4877, 5087, 5443, 5531, 5683, 5923, 6221, 6659, 6791, 6997, 7393, 7603, 8111, 8297, 8641, 8887, 9029, 9377, 9461, 9749

Offset

1,1

Comments

The first prime, 2, is placed at the origin with Cartesian coordinates of (0, 0, 0) and the second prime, 3, is placed at (1, 0, 0). The m-th prime (m >= 3) is placed by moving one unit forward in the direction from the (m-2)-th prime to the (m-1)-th prime, if the next prime is not a twin prime of the current one; otherwise, by turning 90 degrees counterclockwise and moving one unit forward. When it comes to a spot already occupied by another number, the prime is moved up one layer above the number.

Links

Table of n, a(n) for n=1..45.

Example

First layer starts from 2 and second layer from 73.
  59<--53<--47<--43<--41
   |                   |
  61   11<---7<---5   37     137<-131<-127<-113<-109<-107
   |    |         |    |      |                        |
  67   13    2--->3   31     139                      103
   |    |              |                               |
  71   17-->19-->23-->29      73-->79-->83-->89-->97->101

Prog

(Python)
from sympy import prime, nextprime
print(2); d1 = 0; L = [0, 0, 0]; L1 = []
for i in range(1, 1501):
    p = prime(i); np = nextprime(p); d = (d1 + 1)%4 if np - p == 2 else d1
    L[0] += 1 if d == 0 else -1 if d == 2 else 0
    L[1] += 1 if d == 1 else -1 if d == 3 else 0
    if L in L1: L[2] += 1; print(np)
    L1.append([L[0], L[1], L[2]]); d1 = d

Crossrefs

Cf. A063826, A136626.

Sequence in context: A276518 A297410 A041647 * A083018 A240174 A295172

Adjacent sequences: A345290 A345291 A345292 * A345294 A345295 A345296

Keyword

nonn

Author

Ya-Ping Lu, Jun 13 2021

Status

approved