# Sequence of Smallest Numbers whose Reciprocal has Period n

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## Theorem

Let $\sequence {s_n}$ be the sequence defined as:

- $s_n$ is the smallest positive integer the decimal expansion of whose reciprocal has a period of $n$

for $n = 0, 1, 2, \ldots$

Then $\sequence {s_n}$ begins:

- $1, 3, 11, 27, 101, 41, 7, 239, 73, 81, 451, \ldots$

This sequence is A003060 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).

## Proof

Demonstrated by inspection and calculation:

\(\displaystyle \frac 1 1\) | \(=\) | \(\displaystyle 1 \cdotp 0\) | hence has a period of $0$ | ||||||||||

\(\displaystyle \frac 1 3\) | \(=\) | \(\displaystyle 0 \cdotp \dot 3\) | |||||||||||

\(\displaystyle \frac 1 {11}\) | \(=\) | \(\displaystyle 0 \cdotp \dot 0 \dot 9\) | |||||||||||

\(\displaystyle \frac 1 {27}\) | \(=\) | \(\displaystyle 0 \cdotp \dot 03 \dot 7\) | Period of Reciprocal of 27 is Smallest with Length 3 | ||||||||||

\(\displaystyle \frac 1 {101}\) | \(=\) | \(\displaystyle 0 \cdotp \dot 009 \dot 9\) | |||||||||||

\(\displaystyle \frac 1 {41}\) | \(=\) | \(\displaystyle 0 \cdotp \dot 0243 \dot 9\) | |||||||||||

\(\displaystyle \frac 1 7\) | \(=\) | \(\displaystyle 0 \cdotp \dot 14285 \dot 7\) | Period of Reciprocal of 7 is of Maximal Length | ||||||||||

\(\displaystyle \frac 1 {239}\) | \(=\) | \(\displaystyle 0 \cdotp \dot 00418 \, 4 \dot 1\) | |||||||||||

\(\displaystyle \frac 1 {73}\) | \(=\) | \(\displaystyle 0 \cdotp \dot 01369 \, 86 \dot 3\) | |||||||||||

\(\displaystyle \frac 1 {81}\) | \(=\) | \(\displaystyle 0 \cdotp \dot 01234 \, 567 \dot 9\) |

## Sources

- 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): $27$