Problems & Puzzles: Puzzles

Puzzle 173. Ruth-Aaron Triplets

For sure you know the nice story about the pair of consecutive numbers 714 & 715:

In 1974 Hank Aaron got his homerun number 715 eclipsing the Babe Ruth's 1935 record of 714 homeruns. The same year Carl Pomerance noticed that product of 714 & 715 was also the product of the first seven primes.

714x715 = 2x3x5x7x11x13x17

A student of Pomerance also observed that the sum of the prime factors of 714 is equal to the sum of the prime factors of 715

714 = 2x3x7x17, 715 = 5x11x13, 2+3+7+17 = 29 = 5+11+13

Pomerance named pairs like 714-75 Ruth-Aaron pairs, and calculated all the pairs below 20,000.

He also conjectured that this kind of pairs occurred infinitely often, but have no idea of how to prove this when he published this in the JRM.

One week after the publication Paul Erdos got the proof of the infinitude of Ruth Aaron pairs.

See: 1, 2 

Now an ugly issue: what happens if in the prime decomposition of the numbers involved in the Ruth Aaron pairs, one or several of these prime are powered? Will you take all the prime factors involved (with repetitions) or will you take only the distinct primes involved (without repetition)?

As a matter of fact you can do it in one way or another, with the result that you will generate two kind of sequences: Ruth-Aaron pairs, prime factors a) with repetition, A039752 b) without repetition, A006145

Here we will deal with the Pomerance's student's observation about 714 & 715. In particular we will ask for Ruth-Aaron triplets, that is to say three consecutive numbers, n, n+1 & n+2 such that the sum of the prime factors of each number adds up to the same quantity a) without repetition or b) with repetition.

For my surprise Ruth-Aaron triplets exist!

The first example, prime factors with repetition, is the triplet 417162, 417163 & 417164:

417162 = 2x3x251x277
417163 = 17x53x463
417164 = 2x2x11x19x499

2+3+251+277 = 17+53+463 = 2+2+11+19+499 = 533

The first example, prime factors without repetition, is the triplet 89460294, 89460295, 89460296:

89460294 = 2x3x7x11x23x8419
89460295 = 5x4201x4259
89460296 = 2x2x2x31x43x8389

2+3+7+11+23+8419 = 5+4201+4259 = 2+31+43+8389 = 8465


Can you find three more example of each kind?


Joe K. Crump has developed an approach to generate Ruth-Aaron pairs. If his approach can be extended to Ruth-Aaron triplets, Joe has the solution to this puzzle, no matter if the solutions involve a kind of large numbers. See his page.



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