8128

Number
$8128$ (eight thousand, one hundred and twenty-eight) is:


 * $2^6 \times 127$


 * The $4$th perfect number after $6$, $28$, $496$:
 * $8128 = 1 + 2 + 4 + 8 + 16 + 32 + 64 + 127 + 254 + 508 + 1016 + 2032 + 4064 = \map \sigma {8128} - 8128 = 64 \times 127 = 2^{7 - 1} \paren {2^7 - 1}$


 * The $64$th hexagonal number after $1$, $6$, $15$, $28$, $45$, $66$, $91$, $\ldots$, $6903$, $7140$, $7381$, $7626$, $7875$:
 * $8128 = \ds \sum_{k \mathop = 1}^{64} \paren {4 k - 3} = 64 \paren {2 \times 64 - 1}$


 * The $127$th triangular number after $1$, $3$, $6$, $10$, $15$, $\ldots$, $7503$, $7626$, $7750$, $7875$, $8001$:
 * $8128 = \ds \sum_{k \mathop = 1}^{127} k = \dfrac {127 \times \paren {127 + 1} } 2$