After Hank Aaron broke Babe Ruth’s home run record of 714, Carl Pomerance noticed that the prime factors of 714 and 715 both add up to 29. Since then, a pair of integers and is called a Ruth-Aaron pair if the prime factors add up to the same number, and individually is called a Ruth-Aaron number and so is .
Given the factorizations of the numbers
where the and are all distinct primes, and the and are positive integers (not necessarily distinct), and is the number of distinct prime factors function, it becomes apparent that there are at least two different ways to sum up the prime factors.
The most obvious way is to simply test
in which case the first few Ruth-Aaron pairs are (5, 6), (24, 25), (49, 50), (77, 78), (104, 105), (153, 154), (369, 370), (492, 493), (714, 715), … (see A006145 in the OEIS).
The second way is to count repeated prime factors as they occur (e.g., the sum of prime factors of is 12). Thus the test becomes
in which case the first few Ruth-Aaron pairs are (5, 6), (8, 9), (15, 16), (77, 78), (125, 126), (714, 715), (948, 949), … (see A039752 in the OEIS).
The pairs (5, 6), (77, 78), (714, 715), … work under either definition, and it can be observed that , where is the Möbius function (at least for ).
|Date of creation||2013-03-22 16:07:09|
|Last modified on||2013-03-22 16:07:09|
|Last modified by||CompositeFan (12809)|
|Synonym||Ruth Aaron pair|
|Synonym||Aaron Ruth pair|