561

Number
$561$ (five hundred and sixty-one) is:


 * $3 \times 11 \times 17$


 * The $1$st Carmichael number:
 * $\forall a \in \Z: a \perp 561: a^{560} \equiv 1 \pmod {561}$


 * The $2$nd Poulet number after $341$:
 * $2^{561} \equiv 2 \pmod {561}$: $561 = 3 \times 11 \times 17$


 * The $4$th Fermat pseudoprime to base $5$ after $4$, $124$, $217$:
 * $5^{561} \equiv 5 \pmod {561}$


 * The $7$th Fermat pseudoprime to base $4$ after $15$, $85$, $91$, $341$, $435$, $451$:
 * $4^{561} \equiv 4 \pmod {561}$


 * The $17$th hexagonal number after $1$, $6$, $15$, $28$, $45$, $66$, $91$, $120$, $153$, $190$, $231$, $276$, $325$, $378$, $435$, $496$:
 * $\ds 561 = \sum_{k \mathop = 1}^{17} \paren {4 k - 3} = 17 \paren {2 \times 17 - 1}$


 * The $33$rd triangular number after $1$, $3$, $6$, $10$, $15$, $\ldots$, $325$, $351$, $378$, $406$, $435$, $465$, $496$, $528$:
 * $\ds 561 = \sum_{k \mathop = 1}^{33} k = \dfrac {33 \times \paren {33 + 1} } 2$