a(7) is too large to include.
I conjecture that all members of a(n) are members of A054646 and A010120, "Smallest number to give 2^(2n) in a hailstone (3x + 1) sequence" and "Smallest start for a `3x+1' sequence containing 2^n".
Table of n, a(n) for n=1..6.
a(n) = ((2^(3^(n-1)+n)-3^n+2^n))/3^n.
For n = 3, the reduced Collatz sequence k = (3*k+1)/2 is 151, 227, 341, 512.
Cf. A054646 and A010120.
Sequence in context: A157578 A137802 A100203 * A189248 A209393 A095225
Adjacent sequences: A287316 A287317 A287318 * A287320 A287321 A287322
Joe Slater, May 23 2017