360

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Number

$360$ (three hundred and sixty) is:

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

The $1$st number whose ratio to aliquot sum is $4 : 9$:

$\dfrac {\map {\sigma_1} {360} - 360} {360} = \dfrac 9 4$

where $\sigma_1$ denotes the divisor sum function.

The $13$th highly composite number after $1$, $2$, $4$, $6$, $12$, $24$, $36$, $48$, $60$, $120$, $180$, $240$:

$\map {\sigma_0} {360} = 24$

The $13$th superabundant number after $1$, $2$, $4$, $6$, $12$, $24$, $36$, $48$, $60$, $120$, $180$, $240$:

$\dfrac {\map {\sigma_1} {360} } {360} = \dfrac {1170} {360} = 3 \cdotp 25$

The $16$th positive integer after $64$, $96$, $128$, $144$, $160$, $192$, $216$, $224$, $240$, $256$, $288$, $320$, $324$, $336$, $352$ with $6$ or more prime factors:

$360 = 2 \times 2 \times 2 \times 3 \times 3 \times 5$

The $33$rd highly abundant number after $1$, $2$, $3$, $4$, $6$, $8$, $10$, $\ldots$, $120$, $144$, $168$, $180$, $210$, $216$, $240$, $288$, $300$, $336$:

$\map {\sigma_1} {360} = 1170$

The divisor sum of $120$:

$\map {\sigma_1} {120} = 360 = 3 \times 120$

Arithmetic Functions on $360$

\(\ds \map {\sigma_0} { 360 }\)

\(=\)

\(\ds 24\)

$\sigma_0$ of $360$

\(\ds \map {\sigma_1} { 360 }\)

\(=\)

\(\ds 1170\)

$\sigma_1$ of $360$

Also see

• Ratio of 360 to Aliquot Sum
• $\sigma_1$ of $120$

• Previous  ... Next: Highly Composite Number
• Previous  ... Next: Superabundant Number
• Previous  ... Next: Highly Abundant Number
• Previous  ... Next: Numbers with 6 or more Prime Factors

Historical Note

There are $360$ degrees in a full angle.

This appears to have been first done by Hipparchus of Nicaea.

$360$ is very approximately equal to the number of days in a year, which goes some way towards explaining this division of the circle.

Each of the signs of the Zodiac were divided into $30$ degrees, each one corresponding to approximately one day's travel of the sun around the ecliptic.