8128

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Number

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

$2^6 \times 127$

The $4$th perfect number after $6$, $28$, $496$:

$8128 = 1 + 2 + 4 + 8 + 16 + 32 + 64 + 127 + 254 + 508 + 1016 + 2032 + 4064 = \map {\sigma_1} {8128} - 8128 = 64 \times 127 = 2^{7 - 1} \paren {2^7 - 1}$

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 $1$, $3$, $6$, $10$, $15$, $\ldots$, $7503$, $7626$, $7750$, $7875$, $8001$:

$8128 = \ds \sum_{k \mathop = 1}^{127} k = \dfrac {127 \times \paren {127 + 1} } 2$

Also see

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Sources

• 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $127$
• 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $8128$
• 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $127$
• 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $8128$