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, $170=2\cdot5\cdot17$ and $171=3^2\cdot19$ form a pair since $2+5+17=24$ and $3+3+19=25$ .

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 $10^{13}$ 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

$\begin{array}{lll} 27574665988 &=& 2^2 \cdot 139 \cdot 269 ⋅\cdot 331 \cdot 557\\ 27574665989 &=& 13 \cdot 41 \cdot 191 \cdot 439 \cdot 617\\ 27574665990 &=& 2 \cdot 3 \cdot 5 \cdot 19 \cdot 163 \cdot 449 \cdot 661\,, \end{array}$

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

$\small \left(\begin{array}{c}2957791666084\\2957791666085\end{array}\right)\!\Rightarrow\! \left(\begin{array}{c}121539\\121540\end{array}\right)\!\Rightarrow\! \left(\begin{array}{c}170\\171\end{array}\right)\!\Rightarrow\! \left(\begin{array}{c}24\\25\end{array}\right)\!\Rightarrow\! \left(\begin{array}{c}9\\10\end{array}\right)\!\Rightarrow\! \left(\begin{array}{c}6\\7\end{array}\right).$

You can download a text file (eRAP_upto1e12.txt) of 5.4 MB, containing the first members of the 446139 eRAPs up to $10^{12}$ .

Pictorial representation of remainders (mod 2, 3, ...,11) frequency. For a table of values and more details click here

A graph displaying how many eRAPs are multiples of the primes p from 2 to 71. In black the ideal line 1/p.

Useful links OEIS, Sequence A228126
Carlos Rivera, Puzzle 719. Extension to Ruth-Aaron pairs

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

ABA 24 98 171014018 + 28080708128 aban 20 24 98 + 993833000160 abundant 20 24 1104 + 49232136 Achilles 660534263 1605631112 2005604116 + 28080708128 admirable 20 24 alternating 98 1274 3438 + 985270783 amenable 20 24 1104 + 999393125 apocalyptic 2079 4233 4345 + 28518 arithmetic 20 2079 2255 + 9960047 binomial 20 29938677951 412156185486 brilliant 26123 1572827 9625397 + 886859719 c.octagonal 265225 616225 13213225 + 519926081481 c.triangular 197110 compositorial 24 congruent 20 24 1104 + 9960047 constructible 20 24 170 Cunningham 24 170 23408 + 991645522595 Curzon 98 1274 4233 + 192360321 cyclic 4233 21385 26123 + 9960047 d-powerful 24 56563 78524 + 6275378 de Polignac 182939 247475 1220549 + 99740309 decagonal 255208612 deficient 98 170 1274 + 9960047 dig.balanced 170 2079 197110 + 199977000 Duffinian 98 2255 7625 + 9960047 economical 7625 68479 162810 + 19312223 emirpimes 11592089 17219723 31803559 + 66182693 equidigital 7625 68479 162810 + 19312223 esthetic 98 4345 evil 20 24 170 + 999621955 factorial 24 Friedman 263145 265225 616225 frugal 13213225 13595885 13638976 + 919657469 gapful 170 1274 10620 + 99996379650 happy 1274 13350 13775 + 9960047 Harshad 20 24 1104 + 9998143700 heptagonal 978751 658913710 129621298968 hex 176530075057 highly composite 24 hoax 680350 1730404 3246724 + 96954909 Hogben 21058218111 iban 20 24 170 + 220124 idoneal 24 inconsummate 4233 25592 38180 + 991248 interprime 170 10620 23408 + 99699535 Jordan-Polya 24 junction 1104 7625 20220 + 99979346 katadrome 20 98 Lehmer 32740580041 49497717361 396432377719 928645234777 lucky 2079 14905 35167 + 9472585 Lynch-Bell 24 magic 4969188575 magnanimous 20 98 170 metadrome 24 modest 28518 90243 1220549 + 1608329238 Moran 3438 nialpdrome 20 98 76640 + 976533100 nonagonal 24 5811866875 nude 24 1198224 26642688 + 472113432 O'Halloran 20 oban 20 98 octagonal 7105 54153505 693363221 787190405 odious 98 1104 1274 + 999751103 palindromic 23444432 3686336863 98784948789 802959959208 pancake 2917321 30634879 40270826 panconsummate 20 24 pandigital 709929 89271996 185164700 + 8460391257 pernicious 20 24 98 + 9960047 Perrin 236282 persistent 7269813504 10698534927 18362709504 + 58028163749 plaindrome 24 2255 12568899 1222337999 Poulet 34945 power 265225 616225 13213225 + 519926081481 powerful 265225 616225 13213225 + 519926081481 practical 20 24 1104 + 9952712 prim.abundant 20 1017784 23424674 pronic 20 9808030260 13915387332 573205709712 pseudoperfect 20 24 1104 + 991248 repfigit 1104 repunit 21058218111 Ruth-Aaron 24 13775 430604 + 999738845612 self 20 3438 13775 + 999164932 semiprime 26123 182939 1572827 + 97116947 sliding 20 Smith 2079 976096 1444960 + 96954909 sphenic 170 2255 4233 + 99164165 square 265225 616225 13213225 + 519926081481 super Niven 20 24 20220 super-d 10620 26123 121539 + 9750699 superabundant 24 tau 24 4716 10620 + 998593920 tetrahedral 20 triangular 29938677951 412156185486 tribonacci 24 trimorphic 24 uban 20 98 8066000045 Ulam 4233 61456 71284 + 9695764 unprimeable 4716 5368 13350 + 9952712 untouchable 10620 13350 23408 + 958112 upside-down 59713579315 vampire 8509256892 wasteful 20 24 98 + 9960047 Zuckerman 24 Zumkeller 20 24 1104 + 76640 zygodrome 2255 66661111 336655777