A quality inherent in many mathematicians, especially those working with very good knowledge of basic arithmetical properties. This quality is an extended knowledge of the idiosyncrasy, quirks, and peculiarities of certain whole numbers. It manifests itself in recognition of these numbers for their deeper meanings, for example; a person with this ability will instantly recognize many prime numbers up to very large quantities. The number 220 will be automatically associated with 284, because of an odd relationship (the sum of the integer divisors of each one is equal to the other). 257 is an interesting number, as it can be expressed as 223+1 (or 2^(2^3)+1), and a famous hypothesis held that all numbers of the form 22n (2^(2^n)+1) were prime*. Perhaps an extreme example of this would be the number 1729, which is the smallest integer that can be expressed as the sum of two cubes in two different ways.
This quality really isn't limited to higher mathmeticians though. Most of you, being involved in computers, are intimately familiar with the multiples of 8. Many can probably spool off a number of them.."8, 16, 32, 64, 128, 256, 512, 1024" (MB of RAM...).
*I can't say exactly which theorem this was. I do know however, that Fermat first stated this, but his conjecture was later proved false, as when n=5 a composite number is the result.
No, I am neither a mathmetican (yet), nor am I a particularly talented "friend of the integers". I must credit the book Uncle Petros and Goldbach's Conjecture for giving me the inspiration for this writeup