Sequence of Square Lucky Numbers

Theorem

The sequence of lucky numbers which are also square begins:

$1, 9, 25, 49, 169, 289, 361, 529, \ldots$

This sequence is A031162 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).

Proof

The sequence of lucky numbers begins:

$1, 3, 7, 9, 13, 15, 21, 25, 31, 33, 37, 43, 49, 51, 63, 67, 69, 73, 75, 79, 87, 93, 99, \ldots$

Of these, the square ones can be picked out by inspection.

$\blacksquare$