In mathematics, a Ruth–Aaron pair consists of two consecutive integers (e.g., 714 and 715) for which the sums of the prime factors of each integer are equal:
 714 = 2 × 3 × 7 × 17,
 715 = 5 × 11 × 13,
and
 2 + 3 + 7 + 17 = 5 + 11 + 13 = 29.
There are different variations in the definition, depending on how many times to count primes that appear multiple times in a factorization.
The name was given by Carl Pomerance for Babe Ruth and Hank Aaron, as Ruth's career regularseason home run total was 714, a record which Aaron eclipsed on April 8, 1974, when he hit his 715th career home run. Pomerance was a mathematician at the University of Georgia at the time Aaron (a member of the nearby Atlanta Braves) broke Ruth's record, and the student of one of Pomerance's colleagues noticed that the sums of the prime factors of 714 and 715 were equal.^{[1]}
YouTube Encyclopedic

1/1Views:459

✪ The Number 153 : more here: http://goo.gl/9aKzQ
Transcription
Contents
Examples
If only distinct prime factors are counted, 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), (1682, 1683), (2107, 2108)
(The lesser of each pair is listed in OEIS: A006145).
Counting repeated prime factors (e.g., 8 = 2×2×2 and 9 = 3×3 with 2+2+2 = 3+3), the first few Ruth–Aaron pairs are:
(The lesser of each pair is listed in OEIS: A039752).
The intersection of the two lists begins:
 (5, 6), (77, 78), (714, 715), (5405, 5406).
(The lesser of each pair is listed in OEIS: A039753).
Any Ruth–Aaron pair of squarefree integers belongs to both lists with the same sum of prime factors. The intersection also contains pairs that are not squarefree, for example (7129199, 7129200) = (7×11^{2}×19×443, 2^{4}×3×5^{2}×13×457). Here 7+11+19+443 = 2+3+5+13+457 = 480, and also 7+11+11+19+443 = 2+2+2+2+3+5+5+13+457 = 491.
Density
RuthAaron pairs are sparse (that is, they have density 0). This was conjectured by Nelson et al. in 1974^{[2]} and proven in 1978 by Erdős and Pomerance.^{[3]}
Ruth–Aaron triplets
Ruth–Aaron triplets (overlapping Ruth–Aaron pairs) also exist. The first and possibly the second when counting distinct prime factors:
 89460294 = 2 × 3 × 7 × 11 × 23 × 8419,
 89460295 = 5 × 4201 × 4259,
 89460296 = 2 × 2 × 2 × 31 × 43 × 8389,
 and 2 + 3 + 7 + 11 + 23 + 8419 = 5 + 4201 + 4259 = 2 + 31 + 43 + 8389 = 8465.
 151165960539 = 3 × 11 × 11 × 83 × 2081 × 2411,
 151165960540 = 2 × 2 × 5 × 7 × 293 × 1193 × 3089,
 151165960541 = 23 × 29 × 157 × 359 × 4021,
 and 3 + 11 + 83 + 2081 + 2411 = 2 + 5 + 7 + 293 + 1193 + 3089 = 23 + 29 + 157 + 359 + 4021 = 4589.
The first two Ruth–Aaron triplets when counting repeated prime factors:
 417162 = 2 × 3 × 251 × 277,
 417163 = 17 × 53 × 463,
 417164 = 2 × 2 × 11 × 19 × 499,
 and 2 + 3 + 251 + 277 = 17 + 53 + 463 = 2 + 2 + 11 + 19 + 499 = 533.
 6913943284 = 2 × 2 × 37 × 89 × 101 × 5197,
 6913943285 = 5 × 283 × 1259 × 3881,
 6913943286 = 2 × 3 × 167 × 2549 × 2707,
 and 2 + 2 + 37 + 89 + 101 + 5197 = 5 + 283 + 1259 + 3881 = 2 + 3 + 167 + 2549 + 2707 = 5428.
As of 2006^{[update]} only the 4 above triplets are known.^{[citation needed]}
See also
References
 ^ Aaron Numbers  Numberphile
 ^ Nelson, C.; Penney, D. E.; and Pomerance, C. "714 and 715." J. Recr. Math. 7, 8789, 1974.
 ^ Erdős, P. and Pomerance, C. "On the Largest Prime Factors of n and n+1." Aequationes Mathematicae 17, 311321, 1978.
External links
 Weisstein, Eric W. "RuthAaron pair". MathWorld.
 "Ruth–Aaron Triplets" and "Ruth–Aaron pairs revisited". The prime puzzles & problems connection. Retrieved November 9, 2006.