# 128

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## Number

$128$ (one hundred and twenty-eight) is:

$2^7$

In binary:
$128_{10} = 10 \, 000 \, 000_2$

The $1$st positive integer with $7$ or more prime factors:
$128 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2$

The $2$nd seventh power after $1$:
$128 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2$

The $3$rd positive integer after $64$, $96$ with $6$ or more prime factors:
$128 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \paren {\mathop \times 2}$

The $5$th and last of the known powers of $2$ whose digits are also all powers of $2$:
$1$, $2$, $4$, $8$, $128$

The $7$th power of $2$ after $(1)$, $2$, $4$, $8$, $16$, $32$, $64$:
$128 = 2^7$

The $8$th almost perfect number after $1$, $2$, $4$, $8$, $16$, $32$, $64$:
$\map \sigma {128} = 255 = 2 \times 128 - 1$

The $18$th Zuckerman number after $1$, $2$, $3$, $4$, $5$, $6$, $7$, $8$, $9$, $11$, $12$, $15$, $24$, $36$, $111$, $112$, $115$:
$128 = 8 \times 16 = 8 \times \paren {1 \times 2 \times 8}$

The $19$th powerful number after $1$, $4$, $8$, $9$, $16$, $25$, $27$, $32$, $36$, $49$, $64$, $72$, $81$, $100$, $108$, $121$, $125$

The $31$st and largest positive integer which is not the sum of $1$ or more distinct squares:
$2$, $3$, $6$, $7$, $8$, $11$, $12$, $15$, $18$, $19$, $22$, $23$, $24$, $27$, $28$, $31$, $32$, $33$, $43$, $44$, $47$, $48$, $60$, $67$, $72$, $76$, $92$, $96$, $108$, $112$, $128$