$49$ is a perfect square.

$49$ is also a lucky number.

The best description of lucky numbers that I have come across is from Ivars Peterson's MathTrak blog which I quote from below:

Hunting for prime numbers, those evenly divisible only by themselves and 1, requires a sieve to separate them from the rest. For example, the sieve of Eratosthenes, named for a Greek mathematician of the third century B.C., generates a list of prime numbers by the process of elimination.

To find all prime numbers less than, say, 100, the hunter writes down all the integers from 2 to 100 in order (1 doesn't count as a prime). First, 2 is circled, and all multiples of 2 (4, 6, 8, and so on) are struck from the list. That eliminates composite numbers that have 2 as a factor. The next unmarked number is 3. That number is circled, and all multiples of 3 are crossed out. The number 4 is already crossed out, and its multiples have also been eliminated. Five is the next unmarked integer. The procedure continues in this way until only prime numbers are left on the list. Though the sieving process is slow and tedious, it can be continued to infinity to identify every prime number.

Other types of sieves isolate different sequences of numbers. Around 1955, the mathematician Stanislaw Ulam (1909-1984) identified a particular sequence made up of what he called "lucky numbers," and mathematicians have been playing with them ever since.

Starting with a list of integers, including 1, the first step is to cross out every second number: 2, 4, 6, 8, and so on, leaving only the odd integers. The second integer not crossed out is 3. Cross out every third number not yet eliminated. This gets rid of 5, 11, 17, 23, and so on. The third surviving number from the left is 7; cross out every seventh integer not yet eliminated: 19, 39, ... Now, the fourth number from the beginning is 9. Cross out every ninth number not yet eliminated, starting with 27.

This particular sieving process yields certain numbers that permanently escape getting killed. That's why Ulam called them "lucky." See the table below for a list of lucky numbers less than 200.

1 3 7 9 13 15 21 25 31 33 37 43 49 51 63 67 69 73 75 79 87 93 99 105 111 115 127 129 133 135 141 151 159 163 169 171 189 193 195

These lucky numbers should not be confused with Euler's Lucky Numbers, see http://oeis.org/A014556.

For those that are interested, the $66$ days of this year that are related to a lucky number are:

Lucky Number | Date |

1 | Tuesday 1 January |

3 | Thursday 3 January |

7 | Monday 7 January |

9 | Wednesday 9 January |

13 | Sunday 13 January |

15 | Tuesday 15 January |

21 | Monday 21 January |

25 | Friday 25 January |

31 | Thursday 31 January |

33 | Saturday 2 February |

37 | Wednesday 6 February |

43 | Tuesday 12 February |

49 | Monday 18 February |

51 | Wednesday 20 February |

63 | Monday 4 March |

67 | Friday 8 March |

69 | Sunday 10 March |

73 | Thursday 14 March |

75 | Saturday 16 March |

79 | Wednesday 20 March |

87 | Thursday 28 March |

93 | Wednesday 3 April |

99 | Tuesday 9 April |

105 | Monday 15 April |

111 | Sunday 21 April |

115 | Thursday 25 April |

127 | Tuesday 7 May |

129 | Thursday 9 May |

133 | Monday 13 May |

135 | Wednesday 15 May |

141 | Tuesday 21 May |

151 | Friday 31 May |

159 | Saturday 8 June |

163 | Wednesday 12 June |

169 | Tuesday 18 June |

171 | Thursday 20 June |

189 | Monday 8 July |

193 | Friday 12 July |

195 | Sunday 14 July |

201 | Saturday 20 July |

205 | Wednesday 24 July |

211 | Tuesday 30 July |

219 | Wednesday 7 August |

223 | Sunday 11 August |

231 | Monday 19 August |

235 | Friday 23 August |

237 | Sunday 25 August |

241 | Thursday 29 August |

259 | Monday 16 September |

261 | Wednesday 18 September |

267 | Tuesday 24 September |

273 | Monday 30 September |

283 | Thursday 10 October |

285 | Saturday 12 October |

289 | Wednesday 16 October |

297 | Thursday 24 October |

303 | Wednesday 30 October |

307 | Sunday 3 November |

319 | Friday 15 November |

321 | Sunday 17 November |

327 | Saturday 23 November |

331 | Wednesday 27 November |

339 | Thursday 5 December |

349 | Sunday 15 December |

357 | Monday 23 December |

361 | Friday 27 December |

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