Task

Show that the following remarkable formula gives the sequence of non-square natural numbers:

            n + floor(1/2 + sqrt(n)) 

 Print out the values for    n    in the range   1   to   22
 Show that no squares occur for    n    less than one million


This is sequence   A000037   in the OEIS database.




<?php
	//First Task
	for($i=1;$i<=22;$i++){
		echo($i + floor(1/2 + sqrt($i)) . "\n");
	}
 
	//Second Task
	$found_square=False;
	for($i=1;$i<=1000000;$i++){
		$non_square=$i + floor(1/2 + sqrt($i));
		if(sqrt($non_square)==intval(sqrt($non_square))){
			$found_square=True;
		}
	}
	echo("\n");
	if($found_square){
		echo("Found a square number, so the formula does not always work.");
	} else {
		echo("Up to 1000000, found no square number in the sequence!");
	}
?>