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Ruth-Aaron pairs
Two consecutive number    and    form a Ruth-Aaron pair if they share the same sum of prime factors.

The name is commonly used for the two different families obtained taking into account or not the primes multiplicities.

For example, if only distinct primes are counted, then    is a pair, since    and    and  .

The first pairs of this kind are (5,6), (24,25), (49,50), (77,78), (104,105), (153,154), (369,370), (492,493), (714,715), (1682,1683).

If instead repeated primes are counted,   is a pair since  .

The first pairs of this kind are (5,6), (8,9), (15,16), (77,78), (125,126), (714,715), (948,949), (1330,1331), (1520,1521), (1862,1863).

Clearly if both members of a pair are squarefree, then they belong to both sets.

It is conjectured that there are infinite Ruth-Aaron pairs (since this descends from Schinzel's Hypothesis H), however Carl Pomerance has proved that the sum of the reciprocals of Ruth-Aaron numbers is bounded.

A few Ruth-Aaron triples are known (I searched them up to 1013). The first one, counting distinct primes, is formed by 89460294 = 2 × 3 × 7 × 11 × 23 × 8419, 89460295 = 5 × 4201 × 4259 and 89460296 = 23 × 31 × 43 × 8389. Other such triples start at 151165960539, 3089285427491, 6999761340223, and 7539504384825.

Counting all the prime factors, the first triple is given by 417162 = 2 × 3 × 251 × 277, 417163 = 17 × 53 × 463, and 417164 = 22 × 11 × 19 × 499. Another such triple start at 6913943284.

The first numbers which belong to a Ruth-Aaron pair are 5, 6, 8, 9, 15, 16, 24, 25, 49, 50, 77, 78, 104, 105, 125, 126, 153, 154, 369, 370, 492, 493, 714, 715, 948, 949, 1330, 1331, 1520, 1521, 1682, 1683, 1862, 1863 more terms

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