169

Number

$169$ (one hundred and sixty-nine) is:

$13^2$

Can be expressed as the sum of $n$ non-zero squares for all $n$ from $1$ to $155$.

$1! + 6! + 9! = 363 \, 601$, $3! + 6! + 3! + 6! + 0! + 1! = 1454$, $1! + 4! + 5! + 4! = 169$

The $1$st square number which is the difference between two cubes:

$169 = 8^3 - 7^3$

The $2$nd power of $13$ after $(1)$, $13$:

$169 = 13^2$

The $2$nd centered hexagonal number after $1$ which is also square:

$169 = 8^3 - 7^3 = 13^2$

The $2$nd square after $49$ whose decimal representation can be split into two parts which are each themselves square:

$169 = 13^2$; $16 = 4^2$, $9 = 3^2$

$1$, $9$, $25$, $49$, $169$, $\ldots$

The $8$th centered hexagonal number after $1$, $7$, $19$, $37$, $61$, $91$, $127$:

$169 = 1 + 6 + 12 + 18 + 24 + 30 + 36 + 42 = 8^3 - 7^3$

The $11$th positive integer which cannot be expressed as the sum of a square and a prime:

$1$, $10$, $25$, $34$, $58$, $64$, $85$, $91$, $121$, $130$, $169$, $\ldots$

The $13$th square number after $1$, $4$, $9$, $16$, $25$, $36$, $49$, $64$, $81$, $100$, $121$, $144$:

$169 = 13 \times 13$

The $34$th lucky number:

$1$, $3$, $7$, $9$, $13$, $15$, $21$, $\ldots$, $115$, $127$, $129$, $133$, $135$, $141$, $151$, $159$, $163$, $169$, $\ldots$

The $21$st powerful number after $1$, $4$, $8$, $9$, $16$, $25$, $27$, $32$, $36$, $49$, $64$, $72$, $81$, $100$, $108$, $121$, $125$, $128$, $144$