## Sunday, 10 March 2013

### SUNDAY, 10 MARCH 2013

Today is the $69^{th}$ day of the year

$69 = 3 \times 23$

$69$ is a member of the following two Pythagorean triples $(69, 260, 269)$, $(69, 2380, 2381)$.

$69$ is a Lucky Number, see A000959.
Lucky numbers are defined by a sieve like process. Starting with a list of the Natural Numbers$^*$ delete every every second number noting that $2$ is the next number in the list after the number one. We now have a list that looks like $1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21...$ or all the odd numbers.
We continue the process by deleting every $x^{th}$ number where $x$ is the next number in the list that we haven't previously used which means that we use $3$ next i.e. delete every third number in the list to give us $1, 3, 7, 9, 13, 15, 19, 21...$. The next number remaining in the list that we haven't previously used is $7$. Deleting every seventh number leaves the list as $1, 3, 7, 9, 13, 15, 21, ...$.
The  next number remaining in the list that we haven't previously used is $9$ so we delete every ninth number , then every thirteenth, then every fifteenth and so on. The numbers that remain are the Lucky Numbers.
Although the whole sequence can never be determined it is easy, though laborious, to determine whether a number is a member of the sequence by following the above process until either the number in question is removed from the list or the next round of deletions starts past the current position of the number in question in the list.
As far as I can ascertain there is no simple test for whether a number is Lucky.

$^*$ For this purpose the Natural Numbers are considered to be all the positive integers without zero