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1 The \(3k+1\) Problem
1.1 Theory
1.2 Program
1.2-1 ThreeKPlusOneSequence

### 1 The \(3k+1\) Problem

#### 1.1 Theory

Let \(k \in ℕ\) be a natural number. We consider the sequence \(n(i, k), i \in ℕ,\) with \(n(1, k) = k\) and else \(n(i+1, k) = n(i, k) / 2\) if \(n(i, k)\) is even and \(n(i+1, k) = 3 n(i, k) + 1\) if \(n(i, k)\) is odd.

It is not known whether for any natural number \(k \in ℕ\) there is an \(m \in ℕ\) with \(n(m, k) = 1\).

ThreeKPlusOne provides the function `ThreeKPlusOneSequence` (1.2-1) to explore this for given \(n\). If you really want to know something about this problem, see [Wir98] or http://www.ku.de/mgf/mathematik/lehrstuhlstatistik/team/dr-guenther-wirsching/ for more details (and forget this package).

#### 1.2 Program

In this section we describe the main function of this package.

##### 1.2-1 ThreeKPlusOneSequence

`‣ ThreeKPlusOneSequence`( k[, max] ) | ( function )

This function computes for a natural number k the beginning of the sequence \(n(i, k)\) defined in section 1.1. The sequence stops at the first \(1\) or at \(n(\textit{max}, k)\), if max is given.

    gap> ThreeKPlusOneSequence(101);
    "Sorry, not yet implemented. Wait for Version 84 of the package"

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