260

Number
$260$ (two hundred and sixty) is:


 * $2^2 \times 5 \times 13$


 * The $6$th integer $m$ after $0$, $1$, $2$, $8$, $24$ such that $m^2 = \dbinom n 0 + \dbinom n 1 + \dbinom n 2 + \dbinom n 3$ for integer $n$:
 * $260^2 = \dbinom {74} 0 + \dbinom {74} 1 + \dbinom {74} 2 + \dbinom {74} 3$


 * The magic constant of a magic square of order $8$, after $1$, $(5)$, $15$, $34$, $65$, $111$, $175$:
 * $260 = \displaystyle \dfrac 1 8 \sum_{k \mathop = 1}^{8^2} k = \dfrac {8 \paren {8^2 + 1} } 2$


 * The $13$th second pentagonal number after $2$, $7$, $15$, $26$, $40$, $57$, $77$, $100$, $126$, $155$, $187$, $222$:
 * $260 = \dfrac {13 \paren {3 \times 13 + 1} } 2$


 * The $13$th positive integer after $50$, $65$, $85$, $125$, $130$, $145$, $170$, $185$, $200$, $205$, $221$, $250$ which can be expressed as the sum of two square numbers in two or more different ways:
 * $260 = 16^2 + 2^2 = 14^2 + 8^2$


 * The $24$th noncototient after $10$, $26$, $34$, $50$, $\ldots$, $206$, $218$, $222$, $232$, $244$:
 * $\nexists m \in \Z_{>0}: m - \map \phi m = 260$
 * where $\map \phi m$ denotes the Euler $\phi$ function


 * The $26$th generalized pentagonal number after $1$, $2$, $5$, $7$, $12$, $15$, $\ldots$, $126$, $145$, $176$, $187$, $210$, $222$, $247$:
 * $260 = \dfrac {13 \paren {3 \times 13 + 1} } 2$


 * The $51$st positive integer $n$ such that no factorial of an integer can end with $n$ zeroes.