Reciprocals whose Decimal Expansion contain Equal Numbers of Digits from 0 to 9

Theorem
The following positive integers $p$ have reciprocals whose decimal expansions:
 * $(1): \quad$ have the maximum period, that is: $p - 1$
 * $(2): \quad$ have an equal number, $\dfrac {p - 1} {10}$, of each of the digits from $0$ to $9$:


 * $61, 131,\ldots$

Proof
From Reciprocal of 61:
 * $\dfrac 1 {61} = 0 \cdotp \dot 01639 \, 34426 \, 22950 \, 81967 \, 21311 \, 47540 \, 98360 \, 65573 \, 77049 \, 18032 \, 78688 \, 5245 \dot 9$

From Reciprocal of 131:
 * $\dfrac 1 {131} = 0 \cdotp \dot ... \dot .$