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Problem 277

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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”?

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变种考拉兹序列

由a₁开始根据下列规则生成的整数数列被称为变种考拉兹序列：

a_n+1 = a_n/3，如果a_n能被3整除。这是一个较大的下降，我们记这一步操作为“D”。

a_n+1 = (4a_n + 2)/3，如果a_n除以3余1。这是一个较大的上升，我们记这一步操作为“U”。

a_n+1 = (2a_n - 1)/3，如果a_n除以3余2。这是一个较小的下降，我们记这一步操作为“d”。

一旦有a_n = 1，序列结束。

给定任意正整数，我们可以列出其操作步骤。
例如，若a₁=231，整数数列{a_n}={231,77,51,17,11,7,10,14,9,3,1}对应的操作步骤是”DdDddUUdDD”。

当然，存在其它的整数数列其操作步骤的开头同样是”DdDddUUdDD….”。
例如，若a₁=1004064，则操作步骤是DdDddUUdDDDdUDUUUdDdUUDDDUdDD。
事实上，1004064是在a₁ > 10⁶范围内，操作步骤以DdDddUUdDD开头的最小值。

在a₁ > 10¹⁵范围内，操作步骤以”UDDDUdddDDUDDddDdDddDDUDDdUUDd”开头的最小值是多少？

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