A334752 -id:A334752 - cp's OEIS frontend

A068225 Neighbor in 1-2 direction of numbers arranged as clockwise spiral.

2, 11, 12, 3, 4, 1, 8, 9, 10, 27, 28, 29, 30, 13, 14, 15, 16, 5, 6, 7, 22, 23, 24, 25, 26, 51, 52, 53, 54, 55, 56, 31, 32, 33, 34, 35, 36, 17, 18, 19, 20, 21, 44, 45, 46, 47, 48, 49, 50, 83, 84, 85, 86, 87, 88, 89, 90, 57, 58, 59, 60, 61, 62, 63, 64, 37, 38, 39, 40, 41, 42, 43

Offset: 1

Frank Ellermann, Feb 22 2002

			For example, if the spiral is oriented such that 1 is immediately to the right of 2, as shown below, then a(2) = 11 because 11 is immediately to the right of 2.
  21--22--23--24--25--26
  |                    |
  20  7---8---9---10  27
  |   |            |   |
  19  6   1---2   11  28
  |   |       |    |   |
  18  5---4---3   12  29
  |                |   |
  17--16--15--14--13  30
                       |
  36--35--34--33--32--31

Peter Kagey, Table of n, a(n) for n = 1..10000
Frank Ellermann, Some EIS sequences based on a spiral of natural numbers.
Index entries for sequences that are permutations of the natural numbers.

Cf. A068226 (inverse, left), A334751 (above), A334752 (below).

Cf. A054552 (middle row right), A054567 (middle row left).

A341542 Numbers on the square spiral board that are enclosed by four primes.

Original entry on oeis.org

12, 72, 1152, 1452, 1950, 3672, 5520, 6660, 8232, 10302, 10890, 13218, 15288, 15360, 16062, 18042, 20898, 21018, 23628, 25998, 27918, 32190, 37812, 42018, 42462, 48858, 55818, 57192, 80832, 80910, 83340, 91368, 97848, 98640, 104472, 111492, 117498, 119550

Offset: 1

Ya-Ping Lu, Feb 13 2021

nonn

This sequence is similar to A172294, in which the starting number of the square spiral is 0 instead of 1. For a(n) < 10000000, 4 out of the 782 terms in this sequence, 72, 10302, 415380 and 1624350 are absent in A172294, while 6 out of the 784 terms in A172294, 42, 23562, 83232, 205662, 5805690 and 7019850 are absent in this sequence.
Conjecture: This sequence is infinite. If the conjecture holds, then the twin prime conjecture is true.
The 4 neighbors of n in the spiral are A068225, A068226, A334751, and A334752. - Kevin Ryde, Feb 13 2021

Cf. A341541, A341672, A172294, A068225, A068226, A334751, and A334752.

from sympy import isprime
from math import sqrt, ceil
m, m_max = 2, 1000000
while m <= m_max:
    L = [0, 0, 0, 0]
    n = int(ceil((sqrt(m) + 1.0)/2.0))
    z1 = 4*n*n - 12*n + 10
    z2 = 4*n*n - 10*n + 7
    z3 = 4*n*n - 8*n + 5
    z4 = 4*n*n - 6*n + 3
    z5 = 4*n*n - 4*n + 1
    if m > z1 and m < z2: L = [m + 1, m - 8*n + 15, m - 1, m + 8*n - 7]
    elif m > z2 and m < z3: L = [m + 8*n - 5, m + 1, m - 8*n + 13, m - 1]
    elif m > z3 and m < z4: L = [m - 1, m + 8*n - 3, m + 1, m - 8*n + 11]
    elif m > z4 and m < z5: L = [m - 8*n + 9, m - 1, m + 8*n - 1, m + 1]
    if isprime(L[0]) == 1 and isprime(L[1]) == 1 and isprime(L[2]) == 1 and isprime(L[3]) == 1: print(m)
    m += 2

A068226 Neighbor in 2-1 direction of numbers arranged as clockwise spiral.

Original entry on oeis.org

6, 1, 4, 5, 18, 19, 20, 7, 8, 9, 2, 3, 14, 15, 16, 17, 38, 39, 40, 41, 42, 21, 22, 23, 24, 25, 10, 11, 12, 13, 32, 33, 34, 35, 36, 37, 66, 67, 68, 69, 70, 71, 72, 43, 44, 45, 46, 47, 48, 49, 26, 27, 28, 29, 30, 31, 58, 59, 60, 61, 62, 63, 64, 65, 102, 103, 104, 105, 106, 107

Offset: 1

Frank Ellermann, Feb 22 2002

Any offset (x,y) in this spiral is a permutation of the natural numbers.

			For example, if the spiral is oriented such that 1 is immediately to the left of 2, as shown below, then a(1) = 6 because 6 is immediately to the left of 1.
  21--22--23--24--25--26
  |                    |
  20  7---8---9---10  27
  |   |            |   |
  19  6   1---2   11  28
  |   |       |    |   |
  18  5---4---3   12  29
  |                |   |
  17--16--15--14--13  30
                       |
  36--35--34--33--32--31

Peter Kagey, Table of n, a(n) for n = 1..10000
Frank Ellermann, Some EIS sequences based on a spiral of natural numbers.
Index entries for sequences that are permutations of the natural numbers.

Cf. A068225 (inverse, right), A334751 (above), A334752 (below).

A334751 a(n) is the number immediately above n in a clockwise square spiral of the positive integers with the first step to the right.

Original entry on oeis.org

8, 9, 2, 1, 6, 7, 22, 23, 24, 25, 10, 11, 12, 3, 4, 5, 18, 19, 20, 21, 44, 45, 46, 47, 48, 49, 26, 27, 28, 29, 30, 13, 14, 15, 16, 17, 38, 39, 40, 41, 42, 43, 74, 75, 76, 77, 78, 79, 80, 81, 50, 51, 52, 53, 54, 55, 56, 31, 32, 33, 34, 35, 36, 37, 66, 67, 68, 69

Offset: 1

Peter Kagey, May 10 2020

nonn

Equivalently the number immediately below n in a counterclockwise spiral of the positive integers with the first step to the right.

This is a permutation of the positive integers. A334752 is the inverse permutation.

			For n = 1, a(1) = 8 because 8 is immediately above one in the clockwise square spiral with first step to the right:
  21--22--23--24--25--26
  |                    |
  20  7---8---9---10  27
  |   |            |   |
  19  6   1---2   11  28
  |   |       |    |   |
  18  5---4---3   12  29
  |                |   |
  17--16--15--14--13  30
                       |
  36--35--34--33--32--31

Cf. A068225 (right), A068226 (left), A334752 (below).

Cf. A174344 (x-coordinate), A268038 (y-coordinate).

cp's OEIS Frontend

A068225 Neighbor in 1-2 direction of numbers arranged as clockwise spiral.

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A341542 Numbers on the square spiral board that are enclosed by four primes.

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A068226 Neighbor in 2-1 direction of numbers arranged as clockwise spiral.

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A334751 a(n) is the number immediately above n in a clockwise square spiral of the positive integers with the first step to the right.

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Crossrefs