---
id: 5900f4811000cf542c50ff94
title: 'Problem 277: A Modified Collatz sequence'
challengeType: 5
forumTopicId: 301927
dashedName: problem-277-a-modified-collatz-sequence
---

# --description--

A modified Collatz sequence of integers is obtained from a starting value $a_1$ in the following way:

$a_{n + 1} = \frac{a_n}{3}$ if $a_n$ is divisible by 3. We shall denote this as a large downward step, "D".

$a_{n + 1} = \frac{4a_n + 2}{3}$ if $a_n$ divided by 3 gives a remainder of 1. We shall denote this as an upward step, "U".

$a_{n + 1} = \frac{2a_n - 1}{3}$ 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 $a_1 = 231$, 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 $a_1 = 1004064$, then the sequence is DdDddUUdDDDdUDUUUdDdUUDDDUdDD.

In fact, 1004064 is the smallest possible $a_1 > {10}^6$ that begins with the sequence DdDddUUdDD.

What is the smallest $a_1 > {10}^{15}$ that begins with the sequence "UDDDUdddDDUDDddDdDddDDUDDdUUDd"?

# --hints--

`modifiedCollatzSequence()` should return `1125977393124310`.

```js
assert.strictEqual(modifiedCollatzSequence(), 1125977393124310);
```

# --seed--

## --seed-contents--

```js
function modifiedCollatzSequence() {

  return true;
}

modifiedCollatzSequence();
```

# --solutions--

```js
// solution required
```