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A345293
a(n) is the first number on the n-th layer in a layered square spiral of primes.
0
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.
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
Sequence in context: A276518 A297410 A041647 * A083018 A240174 A295172
KEYWORD
nonn
AUTHOR
Ya-Ping Lu, Jun 13 2021
STATUS
approved