## Monday, 18 February 2013

### MONDAY, 18 FEBRUARY 2013

Today is $49^{th}$ day of the year.

$49$ 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