## Tuesday, 27 December 2011

### Tuesday 27 December

Today is the 361 day of the year.
361 = 19^2, so it is a perfect square, the square of a prime and semiprime.
361 = 105 + 120 + 136 which are the 14th, 15th and 16th triangular numbers.
If one removes the last digit this leaves 36 which is also square.

List all the integers starting at 1 then repeatedly apply the following rules;
Determine the number, n, for this iteration as follows:
If this is the first iteration then the n = 2
otherwise
n is the smallest number remaining in the list that is larger than the previous iterations number.
Delete every nth number in the list.
Thus:
Start with 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27,
The first time iteration has n =2 by definition thus every second number is deleted.
This leaves the odd numbers. 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27
The next iteration removes every third number because 3 is the smallest number in the list that is larger than 2.
This leaves 1, 3, 7, 9, 13, 15, 19, 21, 25, 27
The next iteration removes every seventh number because 7 is the smallest number in the list larger than 3.
This leaves 1, 3, 7, 9, 13, 15, 21, 25, 27
The next iteration removes every ninth number because ....
The resulting sequence is the sequence of lucky numbers. 361 is the 86th number in this sequence.