# 60

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## Number

$60$ (sixty) is:

$2^2 \times 3 \times 5$

The $2$nd unitary perfect number after $6$:
$60 = 1 + 3 + 4 + 5 + 12 + 15 + 20$

The $4$th heptagonal pyramidal number after $1$, $8$, $26$:
$60 = 1 + 7 + 18 + 34 = \dfrac {4 \paren {4 + 1} \paren {5 \times 4 - 2} } 6$

The $5$th special highly composite number after $1$, $2$, $6$, $12$

The smallest positive integer which can be expressed as the sum of $2$ odd primes in $6$ ways:
$60 = 7 + 53 = 13 + 47 = 17 + 43 = 19 + 41 = 23 + 37 = 29 + 31$

The $9$th highly composite number after $1$, $2$, $4$, $6$, $12$, $24$, $36$, $48$:
$\map {\sigma_0} {60} = 12$

The $9$th superabundant number after $1$, $2$, $4$, $6$, $12$, $24$, $36$, $48$:
$\dfrac {\map {\sigma_1} {60} } {60} = \dfrac {168} {60} = 2 \cdotp 8$

The $11$th positive integer $n$ after $5$, $11$, $17$, $23$, $29$, $30$, $36$, $42$, $48$, $54$ such that no factorial of an integer can end with $n$ zeroes

The $14$th semiperfect number after $6$, $12$, $18$, $20$, $24$, $28$, $30$, $36$, $40$, $42$, $48$, $54$, $56$:
$60 = 10 + 20 + 30$

The $17$th highly abundant number after $1$, $2$, $3$, $4$, $6$, $8$, $10$, $12$, $16$, $18$, $20$, $24$, $30$, $36$, $42$, $48$:
$\map {\sigma_1} {60} = 168$

The $19$th of $21$ integers which can be represented as the sum of two primes in the maximum number of ways
$1$, $2$, $3$, $4$, $5$, $6$, $7$, $8$, $10$, $12$, $14$, $16$, $18$, $24$, $30$, $36$, $42$, $48$, $60$, $\ldots$

The $23$rd positive integer which is not the sum of $1$ or more distinct squares:
$2$, $3$, $6$, $7$, $8$, $11$, $12$, $15$, $18$, $19$, $22$, $23$, $24$, $27$, $28$, $31$, $32$, $33$, $43$, $44$, $47$, $48$, $60$, $\ldots$

The $35$th positive integer after $2$, $3$, $4$, $7$, $8$, $\ldots$, $44$, $45$, $46$, $49$, $50$, $54$, $55$, $59$ which cannot be expressed as the sum of distinct pentagonal numbers

The base of the sexagesimal system

The number of degrees in an internal angle of an equilateral triangle

### Arithmetic Functions on $60$

 $\ds \map {\sigma_0} { 60 }$ $=$ $\ds 12$ $\sigma_0$ of $60$ $\ds \map \phi { 60 }$ $=$ $\ds 16$ $\phi$ of $60$ $\ds \map {\sigma_1} { 60 }$ $=$ $\ds 168$ $\sigma_1$ of $60$

## Historical Note

There are $60$ seconds in a minute.

There are $60$ minutes of time in an hour, and $60$ minutes of arc in a degree.

These divisions date back to the time of the Babylonians, who used a $60$-based number system for their mathematics and astronomy.