I found the problem here: http://www.artofproblemsolving.com/community/c7h1314296_2x15_is_divisible_by_x
$\large a_1 = 3$
$\large a_{n+1} = 2^{a_n -1} +5$
Prove:
$\large a_n|a_{n+1}$
Initially I tried inducting but found no way to make the statement carry on to the next recurrence.
This problem seems fairly interesting and I wonder if it can be done using elementary means.
The first four values of $a_n$ are as follows:
$3, 9, 261,$
$1852673427797059126777135760139006525652319754650249024631321344126610074238981$
Mathematica verifies the divisibility of $a_4$ by $a_3$, so the statement does appear to hold water.