65

Number
$65$ (sixty-five) is:


 * $5 \times 13$


 * The $1$st rare number:
 * $65 + 56 = 11^2$, $65 - 56 = 3^2$


 * 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 $62$, $63$:
 * $\nexists n \in \Z_{>0}: n = 65 \times \map {s_{10} } n$


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


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


 * The magic constant of a magic square of order $5$, after $1$, $(5)$, $15$, $34$:
 * $65 = \ds \dfrac 1 5 \sum_{k \mathop = 1}^{5^2} k = \dfrac {5 \paren {5^2 + 1} } 2$


 * 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 $9$th positive integer $n$ after $0$, $1$, $5$, $25$, $29$, $41$, $49$, $61$ such that the Fibonacci number $F_n$ ends in $n$


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


 * 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 $39$th (strictly) positive integer after $1$, $2$, $3$, $\ldots$, $37$, $38$, $42$, $43$, $44$, $48$, $49$, $50$, $54$, $55$, $60$, $61$ which cannot be expressed as the sum of distinct primes of the form $6 n - 1$


 * 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