A Ruth-Aaron pair is a pair of consecutive numbers 
such that the sums of the prime
factors of 
and 
are equal. They are so named because they were inspired by the pair (714, 715) corresponding
to Hank Aaron's record-breaking 715th home run on April 8, 1974, breaking Babe Ruth's
earlier record of 714 (Pomerance 2002; Hoffman 1998, pp. 179-181). These have
the factorizations
 |  |  |
(1)
|
 |  |  |
(2)
|
and 
.
If multiplicities are not counted (so that a factor of 
counts only a single 2), then the first few 
s giving Ruth-Aaron pairs are 5, 24, 49, 77, 104, 153, 369,
492, 714, 1682, ... (OEIS A006145), corresponding
to the sums 5, 5, 7, 18, 15, 20, 44, 46, 29, ... (OEIS A006146).
If multiplicities are counted (so that a factor of 
counts as 
, then the first few 
s giving Ruth-Aaron pairs are 5, 8, 15, 77, 125, 714, 948,
... (OEIS A039752), corresponding to the sums
5, 6, 8, 18, 15, 29, 86, ... (OEIS A054378).
The numbers of such 
less than 
,
2, ... are 2, 4, 7, 20, 57, 149, 523, ... (OEIS A101805).
Nelson et al. (1974) showed that a conjecture of Schinzel on simultaneous prime values of polynomials known as "Schinzel's Hypothesis H" would imply the existence of infinitely many Ruth-Aaron pairs. This conjecture remains open (Pomerance 2002), despite an erroneous claimed proof by Erdős claimed by Hoffman (1998, pp. 180-181).
Nelson et al. (1974) also conjectured that Ruth-Aaron pairs were sparse (i.e., have density 0), a conjecture subsequently proved by Erdős and Pomerance (1978),
who showed that if a Ruth-Aaron number is defined as a number 
such that 
where 
is the sum of prime factors of 
taken with multiplicity, then the number of Ruth-Aaron numbers
up to 
is
 |
(3)
|
which can be improved to 
. Pomerance (2002) subsequently improved this bound
to
 |
(4)
|
thus establishing the fact that the sum of the reciprocals of the Ruth-Aaron numbers is bounded. In fact,
 |
(5)
|
and 
." Aeq. Math. 17, 311-321, 1978.Hoffman,
P.