A Ruth-Aaron pair is a pair of consecutive positive integers 
such that the sums of 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 (Hoffman 1998, pp. 179-181;
Pomerance 2002, pp. 567-569). These have the factorizations
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(1)
|
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(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 
for 
, 2, ... are 2, 4, 7, 20, 57, 149, 523, ... (OEIS
A101805).
Nelson et al. (1974, pp. 87-88) showed that Schinzel's hypothesis H on simultaneous prime values of polynomials would imply the existence of infinitely many Ruth-Aaron pairs when multiplicities are counted. The infinitude conjecture remains open (Pomerance 2002, p. 569), despite an erroneous report by Hoffman (1998, pp. 180-181) of a proof by Erdős.
Nelson et al. (1974, pp. 87-88) also conjectured that Ruth-Aaron pairs with multiplicities counted were sparse (i.e., have
natural 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
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(3)
|
which can be improved to 
. Pomerance (2002, p. 570)
subsequently improved this bound to
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(4)
|
thus establishing the fact that the sum of the reciprocals of the Ruth-Aaron numbers is bounded. Numerically,
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(5)
|
and 
." Aeq. Math. 17, 311-321, 1978.Hoffman,
P.