Problems & Puzzles: Puzzles

 Puzzle 615. Abs[R(n/3)-n]<=1 Claudio Meller found that 783742161 and all the numbers of the form (7425)n741, for n=>0, have the following property: R(n/3)-n=1....(*) Example: 741 is such that 741/3= 247, & R(247)=742... I tried to get solutions to this more general property: Abs[R(n/3)-n]<=1.................(**) I only got one more solution to (**), other than the already reported by Meller, for n=42. Q. Can you get other solutions to (**)?

Contributions came from Emmanuel Vantieghem

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Emmanuel wrote:

(7837421625)n 783742161 is a solution for (**). I could not find others but the known ones.

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Wow!... this means that the "singular/isolated" solution reported by Meller (783742161) is in fact the smallest solution of a set of infinite solutions...  As you can see the repeating part (7837421625) is a kind of similar to the extreme/fixed part (783742161). This might have been the clue to get this infinite family of solutions?... I will ask to Emmanuel.(CR)

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This is the comment from Emmanuel to my question:

There is no secret behind the extra solutions I found for puzzle 615.  I just 'tried' to reproduce the link between the solutions  742  and the numbers  (7425)n 742.  And it was successful (was it the lucky of the dummies ?).

Dummies?.... no, I prefer to name it 'holy ingenuity!'...

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