Two integers n and n+1 such that the sums of their respective prime factors are equal.
This property was first noticed for the pair 714,715 and is so named after Hank Aaron's record-breaking 715th home run which was achieved on April 8th 1974, beating the earlier record of 714 held by Babe Ruth.
The result was noticed by a student of one of Carl Pomerance's colleagues at the University of Georgia. It was Pomerance who gave them the name and wrote about their rarity (a computer search of the first 20,000 integers yields only 26 such pairs) in the Journal of Recreational Mathematics. Pomerance suspected that there was an infinite number of these pairs, despite the fact that the pairs thinned out as they became larger (as primes do). He could not prove this until he received a phone call from Erdös. It became the first of more than 40 joint papers that they produced.
In 1995 Both Aaron and Erdös received honourary degrees from Emory University, and when they both signed a baseball for Pomerance Aaron effectively gained an Erdös number of one.
For reference, the following are the values of n under 20000 which will generate Ruth-Aaron pairs:
5, 8, 15, 77, 125, 714, 948, 1330, 1520, 1862, 2491, 3248, 4185, 4191, 5405, 5560, 5959, 6867, 8280, 8463, 10647, 12351, 14587, 16932, 17080, 18490.
For more details, see Sequence A039752 of the online encycolpedia of integer sequences.