61

Number
$61$ (sixty-one) is:


 * The $18$th prime number, after $2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59$


 * The $2$nd of the $7$th pair of twin primes, with $59$


 * The $5$th Keith number after $14, 19, 28, 47$:
 * $6, 1, 7, 8, 15, 23, 38, 61, \ldots$


 * The $5$th centered hexagonal number after $1, 7, 19, 37$:
 * $61 = 1 + 6 + 12 + 18 + 24 = 5^3 - 4^3$


 * The $6$th of $11$ primes of the form $2 x^2 + 11$:
 * $2 \times 5^2 + 11 = 61$


 * The $5$th of $29$ primes of the form $2 x^2 + 29$:
 * $2 \times 4^2 + 29 = 61$


 * The $3$rd after $21, 29$ of the $5$ $2$-digit positive integers which can occur as a $5$-fold repetition at the end of a square number, for example:
 * $1 \, 318 \, 820 \, 881 = 1 \, 739 \, 288 \, 516 \, 161 \, 616 \, 161$


 * The decimal expansion of its reciprocal has the maximum period, that is: $60$:
 * $1 / 61 = 0 \cdotp \dot 01639 \, 34426 \, 22950 \, 81967 \, 21311 \, 47540 \, 98360 \, 65573 \, 77049 \, 18032 \, 78688 \, 5245 \dot 9$


 * It also contains an equal number ($6$) of each of the digits from $0$ to $9$.


 * It is the $1$st positive integer whose reciprocal has this property.


 * The $1$st of the $1$st ordered quadruple of consecutive integers that have sigma values which are strictly increasing:
 * $\sigma \left({61}\right) = 62$, $\sigma \left({62}\right) = 96$, $\sigma \left({63}\right) = 104$, $\sigma \left({64}\right) = 127$

Also see