65

Number
$65$ (sixty-five) is:


 * $5 \times 13$


 * The $23$rd semiprime:
 * $65 = 5 \times 13$


 * The $5$th octagonal number, after $1, 8, 21, 40$:
 * $65 = 1 + 7 + 13 + 19 + 25 = 5 \left({3 \times 5 - 2}\right)$


 * The $5$th Cullen number after $1, 3, 9, 25$:
 * $65 = 4 \times 2^4 + 1$


 * The $8$th integer after $7, 13, 19, 35, 38, 41, 57$ the decimal representation of whose square can be split into two parts which are each themselves square:
 * $65^2 = 4225; 4 = 2^2, 225 = 15^2$


 * The $2$nd positive integer after $50$ which can be expressed as the sum of two square numbers in two or more different ways:
 * $65 = 8^2 + 1^2 = 7^2 + 4^2$


 * and the smallest such which is also the sum of $2$ cube numbers:
 * $65 = 8^2 + 1^2 = 7^2 + 4^2 = 1^3 + 4^3$


 * The $3$rd inconsummate number after $62, 63$:
 * $\nexists n \in \Z_{>0}: n = 65 \times s_{10} \left({n}\right)$


 * The $37$th positive integer after $2, 3, 4, 7, 8, \ldots, 44, 45, 46, 49, 50, 54, 55, 59, 60, 61$ which cannot be expressed as the sum of distinct pentagonal numbers.


 * The magic constant of the order $5$ magic square:
 * $65 = \dfrac {5 \left({5^2 + 1}\right)} 2$


 * In the smallest equilateral triangle with sides of integer length ($112$) which contains a point which is an integer distance from each vertex, the distance from that point to its middle vertex (the other two being $57$ and $73$).

Also see

 * Smallest Equilateral Triangle with Internal Point at Integer Distances from Vertices