28 What are “Lucky Numbers”?

A000959 – OEIS

Lucky numbers are natural numbers generated with a certain sieving principle:

One starts with a list of positive natural numbers. Then one goes through the numbers of the list, beginning with x=2 and crosses out each x-th number. When counting the numbers to be crossed out, the numbers already crossed out are not counted, but only those still in the list. Also when going through the list to get the next x, the deleted ones are skipped.

In the first step, one deletes every second number and thus all even numbers.
In the second step, the number following two in the list is x=3 and every third number is deleted.
In the third step the number following three is x=7 and every seventh number is deleted.
After seven, the number following seven is x=9 and every ninth is deleted.
Then one deletes every 13th, and so on.

Hence the sequence of lucky numbers as all the numbers that are never cancelled:

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, 201, 205, 211, 219, 223, 231, 235, 237, 241, 259, 261, 267, 273, 283, 285, 289, 297, …