561

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Number

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

$3 \times 11 \times 17$

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

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

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