## Monday, April 18, 2011

### Math in baseball: Ruth-Aaron pair

when Hank Aaron's 715th home run beat Babe Ruth's record of 714,
Carl Pomerance noted that

714 = 2*3*7*17 and 715 = 5*11*13

contained the first seven primes and the sums were each 29:
2 + 3 + 7 + 17 = 5 + 11 + 13 = 29

He called n a Ruth-Aaron number if the sum of its prime divisors, counting multiplicity,
was the same as that for n + 1

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.

The number of them that are less than x has been shown by Pomerance to be

O (x(ln lnx)^4 / (ln x)^2)

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

See here

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 http://oeis.org/A039752

(b) without repetition http://oeis.org/A006145

Ruth-Aaron triplets do exist.
That is to say, three consecutive numbers, n, n+1 and 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.

The first example, prime factors with repetition, is the triplet 417162, 417163 and 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

Question:

Can you find three more example of each kind?

Ruth-Aaron pairs revisited