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eRAPs
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.

For example,    and    form a pair since    and  .

The pairs below 10000 are (2, 3), (3, 4), (4, 5), (9, 10), (20, 21), (24, 25), (98, 99), (170, 171), (1104, 1105), (1274, 1275), (2079, 2080), (2255, 2256), (3438, 3439), (4233, 4234), (4345, 4346), (4716, 4717), (5368, 5369), (7105, 7106), and (7625, 7626). more terms

Up to    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

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    is of depth 2, because applying the function sum of prime factors we have    and    is not an eRAP.

Up to    there are 9 eRAPs of depth 5. The smallest one is

You can download a text file (eRAP_upto1e12.txt) of 5.4 MB, containing the first members of the 446139 eRAPs up to  .

ERAPs can also be... (you may click on names or numbers and on + to get more values)