# 60

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## Number

**$60$ (Sixty)** is:

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

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

- The $9$th highly composite number after $1$, $2$, $4$, $6$, $12$, $24$, $36$, $48$:
- $\tau \left({60}\right) = 12$

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

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

- The $9$th superabundant number after $1$, $2$, $4$, $6$, $12$, $24$, $36$, $48$:
- $\dfrac {\sigma \left({60}\right)} {60} = \dfrac {168} {60} = 2 \cdotp 8$

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

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

- The base of the sexagesimal system

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

- The smallest positive integer which is the sum of $2$ odd primes in $6$ different ways

- 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 $11$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 $23$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

### Arithmetic Functions on $60$

\(\displaystyle \map \tau { 60 }\) | \(=\) | \(\displaystyle 12\) | $\tau$ of $60$ | ||||||||||

\(\displaystyle \map \phi { 60 }\) | \(=\) | \(\displaystyle 16\) | $\phi$ of $60$ | ||||||||||

\(\displaystyle \map \sigma { 60 }\) | \(=\) | \(\displaystyle 168\) | $\sigma$ of $60$ |

## Also see

*Previous ... Next*: Numbers not Sum of Distinct Squares*Previous ... Next*: Highly Abundant Number*Previous ... Next*: Smallest Positive Integer which is Sum of 2 Odd Primes in n Ways*Previous ... Next*: Highly Composite Number*Previous ... Next*: Superabundant Number

*Previous ... Next*: Semiperfect Number*Previous ... Next*: Numbers not Expressible as Sum of Distinct Pentagonal Numbers

## 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

- 1986: David Wells:
*Curious and Interesting Numbers*... (previous) ... (next): $60$ - 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): $60$