Problems & Puzzles:
Puzzles
Puzzle 173. RuthAaron 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 71475 RuthAaron
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:
RuthAaron 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
RuthAaron 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 RuthAaron
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
Question:
Can you find three more example of each
kind?
Solution:
Joe K. Crump has developed an approach to
generate RuthAaron pairs. If his approach can be extended to RuthAaron
triplets, Joe has the solution to this puzzle, no matter if the
solutions involve a kind of large numbers. See his page.
***
