496

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Number

$496$ (four hundred and ninety-six) is:

$2^4 \times 31$


The $31$st triangular number after $1$, $3$, $6$, $10$, $15$, $\ldots$, $325$, $351$, $378$, $406$, $435$, $465$:
$496 = \displaystyle \sum_{k \mathop = 1}^{31} k = \dfrac {31 \times \paren {31 + 1} } 2$


The $16$th hexagonal number after $1$, $6$, $15$, $28$, $45$, $66$, $91$, $120$, $153$, $190$, $231$, $276$, $325$, $378$, $435$:
$496 = \displaystyle \sum_{k \mathop = 1}^{16} \paren {4 k - 3} = 16 \paren {2 \times 16 - 1}$


The $3$rd perfect number after $6$, $28$:
$496 = 1 + 2 + 4 + 8 + 16 + 31 + 62 + 124 + 248 = 31 \times 16 = \paren {2^5 - 1} \paren {2^{5 - 1} }$


The $12$th primitive semiperfect number after $6$, $20$, $28$, $88$, $104$, $272$, $304$, $350$, $368$, $464$, $490$:
$496 = 1 + 2 + 4 + 8 + 16 + 31 + 62 + 124 + 248$


The $6$th Ore number after $1$, $6$, $28$, $140$, $270$:
$\dfrac {496 \times \map \tau {496} } {\map \sigma {496} } = 5$


The smallest triangular number which serves as a counterexample to Greenwood's Conjecture:
$496 = T_{31}$ but $496 + 1 = 497 = 7 \times 71$ is not prime


Also see


Sources