A115258 -id:A115258 - cp's OEIS frontend

A136627 For every number n in Ulam's spiral the sequence gives the number of primes around it (number n included).

4, 3, 4, 3, 3, 3, 3, 3, 3, 2, 4, 5, 4, 2, 2, 2, 3, 3, 4, 3, 3, 2, 3, 2, 1, 0, 2, 3, 4, 3, 3, 3, 3, 1, 2, 2, 3, 3, 3, 2, 3, 3, 3, 1, 1, 2, 3, 2, 1, 1, 1, 1, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 2, 1, 3, 3, 3, 1, 2, 3, 5, 4, 4, 3, 2, 0, 1, 2, 2, 1, 1, 1, 1, 1, 2, 2, 2, 1, 2, 2, 1, 0, 2, 2, 4, 3, 3, 1, 0, 1, 1, 2

Offset: 1

Paolo P. Lava and Giorgio Balzarotti, Jan 14 2008

In Ulam's lattice there are 8 numbers around any number. The sequence is similar to A136626 with an increment of 1 for any prime position.

			Numbers around 13 are 3, 12, 29, 30, 31, 32, 33, 14 -> 3, 29, 31 and 13 itself are primes, so a(13)=4.

Cf. A063826, A115258, A136626.

Offset 1 per example and correction for a(32) by Kevin Ryde, Jul 04 2020

A357376 The lowest number on Ulam Spiral for which all numbers in the square which is centered at a(n) and spans n-1 spaces in each cardinal direction are nonprime.

Original entry on oeis.org

1, 26, 1016, 5136, 39639, 203100, 2729736, 32264250, 42119062, 1065799391, 12444190246

Offset: 1

Samuel Harkness, Sep 26 2022

The n-th term creates a (2*n-1) X (2*n-1) square centered at a(n).

			For n=2, test squares with 1 layer of numbers around the center number until a square is enclosed with only nonprimes.
.
   65   64   63   62   61   60   59   58   57
.
   66   37   36   35   34   33   32   31   56
.
   67   38   17   16   15   14   13   30   55
.
   68   39   18    5    4    3   12   29   54
.
   69   40   19    6    1    2   11   28   53
.                              +--------------+
   70   41   20    7    8    9 | 10   27   52 |
.                              |              |
   71   42   21   22   23   24 | 25   26   51 |
.                              |              |
   72   43   44   45   46   47 | 48   49   50 |
.                              +--------------+
   73   74   75   76   77   78   79   80   81
.
The first number tested, 1, does not work because 2, 3, 5 and 7 are in the 1-layered square and are prime. The first square that works is 26 because 10, 27, 52, 25, 26, 51, 48, 49, and 50 are all nonprime.

Samuel Harkness, Illustration of terms 1 through 4
Rémy Sigrist, C++ program

Cf. A000040, A115258, A136626, A225073.

a(8)-a(11) from Rémy Sigrist, Sep 27 2022

A346948 Isolated single primes enclosed by six composites on hexagonal spiral board of odd numbers.

Original entry on oeis.org

211, 257, 277, 331, 509, 563, 587, 647, 653, 673, 683, 709, 751, 757, 839, 853, 919, 983, 997, 1087, 1117, 1123, 1163, 1283, 1433, 1447, 1493, 1531, 1579, 1637, 1733, 1777, 1889, 1913, 1973, 1993, 2179, 2207, 2251, 2273, 2287, 2333, 2357, 2399, 2447, 2467

Offset: 1

Ya-Ping Lu, Aug 08 2021

nonn

It seems that more isolated primes, m, appear in regions 6*k^2-16*k+13 <= m <= 6*k^2-14*k+7 and 6*k^2-10*k+7 <= m <= 6*k^2-8*k+1 than the other 4 regions, where k (>= 1) is the layer number on the hexagonal board, which is illustrated in A345654.
Numbers of prime terms appearing in the 6 regions and 6 arms of a 10000-layer hexagonal board, with the 299970001 odd numbers up to 599940001, are:
               Region               Appearance          Arm         Appearance
----------------------------------  ----------   -----------------  ----------
6*k^2-18*k+15 <= m <= 6*k^2-16*k+9   2681490     m = 6*k^2-16*k+11      692
6*k^2-16*k+13 <= m <= 6*k^2-14*k+7   3192576     m = 6*k^2-14*k+ 9      551
6*k^2-14*k+11 <= m <= 6*k^2-12*k+5   2681571     m = 6*k^2-12*k+ 7      671
6*k^2-12*k+ 9 <= m <= 6*k^2-10*k+3   2681254     m = 6*k^2-10*k+ 5      545
6*k^2-10*k+ 7 <= m <= 6*k^2- 8*k+1   3191045     m = 6*k^2- 8*k+ 3      721
6*k^2- 8*k+ 5 <= m <= 6*k^2- 6*k-1   2680620     m = 6*k^2- 6*k+ 1     1040

			3 is not a term because four of the six neighbors (1, 5, 13, 15, 17 and 19) are primes.
211 is a term because 211 is a prime and all six neighbors (145, 147, 209, 213, 287 and 289) are composites.

Cf. A083577, A090687, A176608, A115258, A341542, A344481, A345654.

from sympy import isprime; from math import sqrt, ceil
def neib(m):
    if m == 1: return [3, 5, 7, 9, 11, 13]
    if m == 3: return [17, 19, 5, 1, 13, 15]
    L = [m for i in range(6)]; n = int(ceil((3+sqrt(6*m + 3))/6)); x=6*n*n; y=12*n
    a0 = x-18*n+15; a1 =x-16*n+11; a2 =x-14*n+9
    a3 = x-y+7; a4 =x-10*n+5; a5 =x-8*n+3; a6 =x-6*n+1
    p = 0 if m==a0 else 1 if m>a0 and ma1 and ma2 and ma3 and ma4 and ma5 and m

A347337 The prime numbers ordered by their distance from the origin of a square spiral which starts at 1. If two or more primes have the same distance from the origin the smaller prime appears first.

Original entry on oeis.org

2, 3, 5, 7, 11, 19, 23, 13, 17, 29, 41, 47, 53, 61, 31, 37, 43, 59, 67, 71, 79, 97, 107, 73, 83, 89, 103, 109, 113, 127, 139, 151, 163, 137, 149, 101, 191, 131, 167, 173, 179, 193, 233, 281, 251, 263, 283, 157, 181, 199, 223, 229, 269, 277, 317, 353, 313, 331, 337, 349, 197, 211, 227, 239

Offset: 1

Scott R. Shannon, Aug 27 2021

nonn

			The square spiral is numbered as follows:
.
  17--16--15--14--13   .
   |               |   .
  18   5---4---3  12   29
   |   |       |   |   |
  19   6   1---2  11   28
   |   |           |   |
  20   7---8---9--10   27
   |                   |
  21--22--23--24--25--26
.
a(1) = 2 as this is the only prime a distance of 1 unit from the origin.
a(2) = 3, a(3) = 5, a(4) = 7. All of these primes are a distance of
sqrt(2) units from the origin, so they are listed with the smaller primes first.
a(6) = 19 as this is 2 units from the origin, the same as a(5) = 11. This is the first prime that appears out of natural order as both 13 and 17 are sqrt(8) units, which is greater than 2 units, from the origin.

Cf. A000040, A063826, A214664, A214665, A136626, A115258, A331027.

cp's OEIS Frontend

A136627 For every number n in Ulam's spiral the sequence gives the number of primes around it (number n included).

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A357376 The lowest number on Ulam Spiral for which all numbers in the square which is centered at a(n) and spans n-1 spaces in each cardinal direction are nonprime.

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A346948 Isolated single primes enclosed by six composites on hexagonal spiral board of odd numbers.

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A347337 The prime numbers ordered by their distance from the origin of a square spiral which starts at 1. If two or more primes have the same distance from the origin the smaller prime appears first.

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