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# Ulam spiral

### From OeisWiki

The **Ulam spiral** is an arrangement of numbers upon which the prime numbers tend to fall on certain diagonals. It may begin on any integer, and may be oriented a few different ways. But the version most commonly presented begins with 1 at the center, and takes its cue for orientation from the March 1964 cover of *Scientific American*, with 2 to the right of 1, and 3 above 2, so that the northern ray reads 1, 4, 15, 34...

## Contents |

## Original Ulam spiral

In the spiral, even numbers are shown in red, odd numbers in black and prime numbers are in **bold**.

A053755: Numbers of the form 4n^{2} + 1A121326: Primes of this form | A054556: Numbers of the form 4n^{2} − 9n + 6A168023: Non-composites of this form | A002378: Numbers of the form n(n + 1)2 is the only even prime | A054554: Numbers of the form 4n^{2} − 10n + 7A073337: Primes of this form | |||||||

A054567: Numbers of the form 4n^{2} − 7n + 4A168025: Non-composites of this form | 65 | 64 | 63 | 62 | 61
| 60 | 59
| 58 | 57 | A054552: Numbers of the form 4n^{2} − 3n + 1A168022: Non-composites of this form |

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 | ||

A054569: Numbers of the form 4n^{2} − 6n + 3A168026: Non-composites of this form | A033951: Numbers of the form 4n^{2} + 3n + 1A168027: Non-composites of this form | A073577: Numbers of the form (2n + 1)^{2} − 2A028871: Primes of this form | A016754: Odd squares, (2n + 1)^{2}No primes of this form. |

Stanislaw Ulam first studied this spiral in 1963, doodling "while sitting through a boring talk."^{[1]} Other starting values can be used and the same clustering of primes along certain diagonals is also observed, such as starting with 41.^{[2]}

## Odd numbers Ulam spiral

In the odd number variant of the **Ulam spiral**, unimpeded by the even numbers, the prime numbers can line up in horizontal and vertical lines. But there are still noticeable diagonal lines of primes, and these primes fall on one such diagonal.

221 | 223 | 225 | 227 | 229 | 231 | 233 | 235 | 237 | 239 | 241 |

219 | 145 | 147 | 149 | 151 | 153 | 155 | 157 | 159 | 161 | 163 |

217 | 143 | 85 | 87 | 89 | 91 | 93 | 95 | 97 | 99 | 165 |

215 | 141 | 83 | 41 | 43 | 45 | 47 | 49 | 51 | 101 | 167 |

213 | 139 | 81 | 39 | 13 | 15 | 17 | 19 | 53 | 103 | 169 |

211 | 137 | 79 | 37 | 11 | 1 | 3 | 21 | 55 | 105 | 171 |

209 | 135 | 77 | 35 | 9 | 7 | 5 | 23 | 57 | 107 | 173 |

207 | 133 | 75 | 33 | 31 | 29 | 27 | 25 | 59 | 109 | 175 |

205 | 131 | 73 | 71 | 69 | 67 | 65 | 63 | 61 | 111 | 177 |

203 | 129 | 127 | 125 | 123 | 121 | 119 | 117 | 115 | 113 | 179 |

201 | 199 | 197 | 195 | 193 | 191 | 189 | 187 | 185 | 183 | 181 |

Note the diagonal of primes of the form (Cf. A090684)

- {7, 31, 71, 127, 199, 647, 967, 1151, 1567, 2311, 2591, 2887, 3527, 4231, 4999, 5407, 6271, 7687, 8191, 11551, 12799, 16927, 19207, 20807, 23327, 25087, 27847, 31751, 34847, 35911, ...}

Note the horizontal line of primes of the form (Cf. A188382)

- {11, 37, 79, 137, 211, 821, 991, 1597, 1831, 2081, 2347, 2927, 3571, 3917, 4657, 5051, 6329, 8779, 9871, 11027, 14197, 14879, 17021, 20101, 21737, 26107, 27967, 28921, 33931, 34981, ...}

Note the vertical line of primes of the form (Cf. A187677)

- {13, 43, 89, 151, 229, 433, 701, 859, 1033, 1223, 1429, 1889, 2143, 2699, 3001, 3319, 4003, 4751, 5563, 7873, 10009, 11173, 11779, 12401, 13693, 17203, 18719, 19501, 21943, 25423, ...}

## Ulam spiral with numbers congruent to 1 or 5 (mod 6)

331 | 335 | 337 | 341 | 343 | 347 | 349 | 353 | 355 | 359 | 361 |

329 | 217 | 221 | 223 | 227 | 229 | 233 | 235 | 239 | 241 | 245 |

325 | 215 | 127 | 131 | 133 | 137 | 139 | 143 | 145 | 149 | 247 |

323 | 211 | 125 | 61 | 65 | 67 | 71 | 73 | 77 | 151 | 251 |

319 | 209 | 121 | 59 | 19 | 23 | 25 | 29 | 79 | 155 | 253 |

317 | 205 | 119 | 55 | 17 | 1 | 5 | 31 | 83 | 157 | 257 |

313 | 203 | 115 | 53 | 13 | 11 | 7 | 35 | 85 | 161 | 259 |

311 | 199 | 113 | 49 | 47 | 43 | 41 | 37 | 89 | 163 | 263 |

307 | 197 | 109 | 107 | 103 | 101 | 97 | 95 | 91 | 167 | 265 |

305 | 193 | 191 | 187 | 185 | 181 | 179 | 175 | 173 | 169 | 269 |

301 | 299 | 295 | 293 | 289 | 287 | 283 | 281 | 277 | 275 | 271 |

## Notes

- ↑ D. Wells,
*Prime Numbers: The Most Mysterious Figures in Math*Hoboken, New Jersey: John Wiley & Sons Inc. (2005) p. 232 - ↑ Ibid., p. 233

## External links

- From BackIssues.com, cover of the
*Scientific American*March 1964 issue - YouTube, Ulam spiral video with digits
- YouTube, Ulam spiral video without digits