Problems & Puzzles: Puzzles

 Puzzle 719 Extension to Ruth Aaron pairs Abhiram R Devesh sent the following puzzle The following puzzle is based on extension of Ruth-Aaron Pairs (let’s call this henceforth as eRAP).  If the sum of prime factors (recurrence of prime factors included) of 2 consecutive numbers is also consecutive, then the pair is an eRAP. eg. n     = 20, prime factors are [2,2,5], Sum of divisors is 9 n+1 = 21, prime factors are [3,7], Sum of divisors is 10 Hence (20, 21) is an eRAP. for an eRAP we define “depth of an eRAP” as the number of levels through which this property holds true for depth=3  Example 1 121539 , 121540 => 170 , 171=>24 , 25 => 9 , 10 sum of prime factors of (121539 , 121540) is (170 , 171) (depth 1) sum of prime factors of (170,171) is (24,25) (depth 2) sum of prime factors of (24,25) is (9,10) (depth 3) Example 2: 295274 , 295275 =>170 , 171 =>24 , 25=>9 , 10  Q1. Are there deeper eRAPs ?  Q2. Are there eRATs (extended Ruth Aaron Triplets) with depth at least 1? Other than the following trivial triplets (2,3,4) and (3,4,5), I could not find any more triplets for n < 107. ____ See http://oeis.org/A228126

Contributions came from Vicente F. Izquierdo, Giovanni Resta and Emmanuel Vantieghem

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Vicente wrote:

Deepness of the pair 121539 y 121540 is 4 not 3 as said by Abhiram because:

121539 , 121540 => 170 , 171=>24 , 25 => 9 , 10 => 6 , 7

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Giovanni wrote:

I finished my search for Puzzle 719 at 10^13.

up to 10^13 there are:

1917636 pairs at level 1
1289  pairs at level 2
1 pair at level 3, i.e., (170,171)
3 pairs at level 4, i.e.,  (121539, 121540), (295274,295275) and (11093225, 11093226).
9 pairs at level 5, whose first members are
2957791666084, 3011043276149, 4016669254756, 6306125953827,
6538721538474, 7501658878220, 7938367737268, 8531992167314,
and 9670214064608.
All these 9 pairs go to the pair (121539, 121540).

up to 10^13 there are also 3 triples, namely:

{27574665988, 27574665989, 27574665990},
{1862179264458, 1862179264459, 1862179264460}, and
{9600314395008, 9600314395009, 9600314395010}.

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Emmanuel wrote:

It was not difficult to find a level 4 solution.  The smallest pair I found was
(6306125953827, 6306125953828). It maps on the pair  (121539,121540). [This is really level 5, as said above]

I do not know if there exist a smaller pair that is the start of a level 4 chain.
The rest of the week I tried to find a level 5 chain, but that was much more difficult.

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