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

Author

Ya-Ping Lu, Jun 13 2021

Keywords

• nonn

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.

Examples

			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
		

Crossrefs

Cf. A063826, A136626.

Programs

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