Ruth-Aaron pair

After Hank Aaron broke Babe Ruth’s home run record of 714, Carl Pomerance noticed that the prime factorsMathworldPlanetmath of 714 and 715 both add up to 29. Since then, a pair of integers n and m=n+1 is called a Ruth-Aaron pair if the prime factors add up to the same number, and individually n is called a Ruth-Aaron number and so is m.

Given the factorizations of the numbers


where the pi and qj are all distinct primes, and the ai and bj are positive integers (not necessarily distinct), and ω(x) 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 72=2332 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 |μ(n)|=|μ(m)|=1, where μ(x) is the Möbius function (at least for n<1777028).

Title Ruth-Aaron pair
Canonical name RuthAaronPair
Date of creation 2013-03-22 16:07:09
Last modified on 2013-03-22 16:07:09
Owner CompositeFan (12809)
Last modified by CompositeFan (12809)
Numerical id 5
Author CompositeFan (12809)
Entry type Definition
Classification msc 11A51
Synonym Ruth Aaron pair
Synonym Aaron-Ruth pair
Synonym Aaron Ruth pair
Defines Ruth-Aaron number