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 $451$

$\blacksquare$


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