A Modified Collatz sequence

A modified Collatz sequence of integers is obtained from a starting value a₁ in the following way:

a_n+1 = a_n/3 if a_n is divisible by 3. We shall denote this as a large downward step, "D".

a_n+1 = (4a_n + 2)/3 if a_n divided by 3 gives a remainder of 1. We shall denote this as an upward step, "U".

a_n+1 = (2a_n - 1)/3 if a_n divided by 3 gives a remainder of 2. We shall denote this as a small downward step, "d".

The sequence terminates when some a_n = 1.

Given any integer, we can list out the sequence of steps. For instance if a₁=231, then the sequence {a_n}={231,77,51,17,11,7,10,14,9,3,1} corresponds to the steps "DdDddUUdDD".

Of course, there are other sequences that begin with that same sequence "DdDddUUdDD....". For instance, if a₁=1004064, then the sequence is DdDddUUdDDDdUDUUUdDdUUDDDUdDD. In fact, 1004064 is the smallest possible a₁ > 10⁶ that begins with the sequence DdDddUUdDD.

What is the smallest a₁ > 10¹⁵ that begins with the sequence "UDDDUdddDDUDDddDdDddDDUDDdUUDd"?

Written by gamwe6 who lives and works in San Francisco building useful things. You should follow him on Twitter