180

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Number

$180$ (one hundred and eighty) is:

$2^2 \times 3^2 \times 5$

The $4$th number after $1$, $9$, $20$ whose square has a divisor sum which is itself square:

$\map {\sigma_1} {180^2} = 341^2$

The $11$th highly composite number after $1$, $2$, $4$, $6$, $12$, $24$, $36$, $48$, $60$, $120$:

$\map \tau {180} = 18$

The $11$th superabundant number after $1$, $2$, $4$, $6$, $12$, $24$, $36$, $48$, $60$, $120$:

$\dfrac {\map {\sigma_1} {180} } {180} = \dfrac {546} {180} = 3 \cdotp 0 \dot 3$

The $26$th highly abundant number after $1$, $2$, $3$, $4$, $6$, $8$, $10$, $12$, $16$, $18$, $20$, $24$, $30$, $36$, $42$, $48$, $60$, $72$, $84$, $90$, $96$, $108$, $120$, $144$, $168$:

$\map {\sigma_1} {180} = 546$

$180^3 = 6^3 + 7^3 + \cdots + 67^3 + 68^3 + 69^3$

The number of degrees in the sum of the internal angles of a triangle

Arithmetic Functions on $180$

\(\ds \map {\sigma_0} { 180 }\)

\(=\)

\(\ds 18\)

$\sigma_0$ of $180$

\(\ds \map {\sigma_1} { 180 }\)

\(=\)

\(\ds 546\)

$\sigma_1$ of $180$

Also see

• Measurement of Straight Angle
• Sum of Angles of Triangle equals Two Right Angles
• Cube of 180 is Sum of Sequence of Consecutive Cubes

• Previous  ... Next: Square Numbers whose Divisor Sum is Square

• Previous  ... Next: Highly Composite Number
• Previous  ... Next: Superabundant Number

• Previous  ... Next: Highly Abundant Number

Historical Note

There are $180 \degrees$ in a straight angle.

Thus the colloquial expression doing a $180$, meaning turning completely around (either literally or metaphorically) to face in the opposite direction, can be found in a variety of contexts.

$180 \fahr$ is the number of degrees Fahrenheit between the melting point ($32 \fahr$) and boiling point ($212 \fahr$) of water.

Sources

• 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $180$
• 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $180$