For a positive integer greater than , define the sum of all pairs: as the sum of all possible pair products, made of distinct integers from to .
If the arithmetic mean of the products is an integer, then is defined as "Soapy".
- is "Soapy" since and there is pair and is an integer
- is not "Soapy" since and there are pairs and is not an integer
Are numbers of the form "Soapy", where ?