Ruth-Aaron Pairs and eRAPs

Today I turned 25593 days old and on Numbers Aplenty I read the following:

With its predecessor 25592, 25593 forms an eRAP, since the sums of their prime factors are consecutive (470 and 471).

There was a link to eRAP that began:

Abhiram R. Devesh proposed an extension of Ruth-Aaron Pairs (thus called eRAP) where two consecutive numbers form a pair if the sums of their prime factors are consecutive. 

So before delving into eRAPs, I'll discuss the RAPs or Ruth-Aaron Pairs. Here is what Numbers Aplenty had to say about these pairs of numbers:

Two consecutive number \(n \) and \(n+1\) 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 \((104, 105) \) is a pair, since \(104=2^3\cdot13 \) and  \(105=3\cdot5\cdot7\) and \(2+13=3+5+7\). 

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, \((125=5^3, 126=2\cdot 3^2\cdot7) \) is a pair since \(5\cdot3 = 2+3\cdot2+7\). 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 square-free, 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 

• 89460296 = 23 × 31 × 43 × 8389. 

Other such triples start at: 

• 151165960539 

• 3089285427491 

• 6999761340223 

• 7539504384825

Counting all the prime factors, the first triple is given by: 

• 417162 = 2 × 3 × 251 × 277 

• 417163 = 17 × 53 × 463 

• 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

This leads us on to eRAPs or extended Ruth_Aaron Pairs defined as consecutive numbers whose respective sums of prime divisors are also consecutive. It seems that repeated prime divisors are included. Here is a list of such pairs up to 25600:

Figure 1: eRAPs between 1 and 25600

Numbers Aplenty goes on to give the following information about eRAPs:

Up to \(10^{13}\)  there are only 5 eRat triples, namely (2, 3, 4), (3, 4, 5), (27574665988, 27574665989, 27574665990), (1862179264458, 1862179264459, 1862179264460), and (9600314395008, 9600314395009, 9600314395010). 

For the smallest nontrivial triple we have: 

\(27574665988 =2^2 \cdot 139 \cdot 269 ⋅\cdot 331 \cdot 557 \)

\(27574665989 =13 \cdot 41 \cdot 191 \cdot 439 \cdot 617 \)

\(27574665990 =2 \cdot 3 \cdot 5 \cdot 19 \cdot 163 \cdot 449 \cdot 661\) 

and the sums of prime factors (with multiplicities) are 1300, 1301, and 1302, respectively. Devesh defines the "depth of an eRAP" as the number of levels through which this property holds true. For example, the pair \((24,25)\) is of depth 2, because applying the function sum of prime factors we have \( (24,25) \Rightarrow (9,10) \Rightarrow (6,7) \)  and \( (6,7) \) is not an eRAP. Up to \(10^{13} \) there are 9 eRAPs of depth 5. The smallest one is 

It should also be noted that these number properties of Ruth-Aaron Pairs and eRAPs are independent of the number base being used as opposed say to d-powerful numbers that can be expressed in terms of the sum of various powers of their digits e.g.\( 3459872 = 3^1 + 4^6 + 5^5 + 9^6 + 8^3 + 7^7 + 2^{21} \). Generally speaking, this property only holds true in base 10. Number properties that are base-independent are intrinsically of more interest to number theorists. Primeness is perhaps the most well-known of these properties.