Problems & Puzzles: Puzzles

 Puzzle 632. [R(n)+1]@n=0 JM Bergot sent the following puzzle: [R(n)+1]@n=0 for n=14 & 37. ¿Is this rare? I made a quick verification and found two things: a) There is a family with infinite terms that satisfy [R(n)+1]@n=0: n= 3(9)m7; for m=0,1,2,... b) Out of this family I found the following three isolated solutions: 14 (Bergot), 122537, 1172839, ?,... Q1. Can you find more solutions out of the family 3(9)m7 Q2. Is there another infinite family as 3(9)m7 satisfying [R(n)+1]@n=0?

Contributions came from Luke Pebody & Emmanuel Vantieghem

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Luke wrote:

The only solutions are:
R(n) = 2n-1: 1, 37, 3(9)m7
R(n) = 3n-1: 14
R(n) = 6n-1: 122537, 122537(74622537)m
R(n) = 8n-1: 1172839, 1172839(6172839)m

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Emmanuel wrote:

I could find two more infinite families of solutions :
n = 122537 (74622537)k  (k = 0, 1, 2, ...).
and
n = 117839 (6172839)k  (k = 0, 1, 2, ...).

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