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 $60$


Also see


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.


Sources