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:
\(\ds \frac 1 1\) | \(=\) | \(\ds 1 \cdotp 0\) | hence has a period of $0$ | |||||||||||
\(\ds \frac 1 3\) | \(=\) | \(\ds 0 \cdotp \dot 3\) | Reciprocal of $3$ | |||||||||||
\(\ds \frac 1 {11}\) | \(=\) | \(\ds 0 \cdotp \dot 0 \dot 9\) | Reciprocal of $11$ | |||||||||||
\(\ds \frac 1 {27}\) | \(=\) | \(\ds 0 \cdotp \dot 03 \dot 7\) | Reciprocal of $27$ | |||||||||||
\(\ds \frac 1 {101}\) | \(=\) | \(\ds 0 \cdotp \dot 009 \dot 9\) | Reciprocal of $101$ | |||||||||||
\(\ds \frac 1 {41}\) | \(=\) | \(\ds 0 \cdotp \dot 0243 \dot 9\) | Reciprocal of $41$ | |||||||||||
\(\ds \frac 1 7\) | \(=\) | \(\ds 0 \cdotp \dot 14285 \dot 7\) | Reciprocal of $7$ | |||||||||||
\(\ds \frac 1 {239}\) | \(=\) | \(\ds 0 \cdotp \dot 00418 \, 4 \dot 1\) | Reciprocal of $239$ | |||||||||||
\(\ds \frac 1 {73}\) | \(=\) | \(\ds 0 \cdotp \dot 01369 \, 86 \dot 3\) | Reciprocal of $73$ | |||||||||||
\(\ds \frac 1 {81}\) | \(=\) | \(\ds 0 \cdotp \dot 01234 \, 567 \dot 9\) | Reciprocal of $81$ | |||||||||||
\(\ds \frac 1 {451}\) | \(=\) | \(\ds 0 \cdotp \dot 00221 \, 7294 \dot 9\) | Reciprocal of $451$ |
$\blacksquare$
Sources
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $27$