has the property that if one adds it to double its square, it yields a perfect square. In other words for natural numbers :
There are four such . If is the smallest , what is the second smallest ?
What is the largest integer for which is a perfect square?
can be written as a sum of consecutive perfect squares and also consecutive non-zero perfect squares:
What is the next number with this property?
For how many positive integers is a perfect square?
Find the sum of all positive integers such that can be expressed as sums of four factorials (of positive integers).
Details and assumptions
The number , read as n factorial, is equal to the product of all positive integers less than or equal to . For example, .
The factorials do not have to be distinct. For example, counts, because it equals .