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 $37$th (strictly) positive integer after $1$, $2$, $3$, $\ldots$, $37$, $38$, $42$, $43$, $44$, $48$, $49$, $50$, $54$, $55$ which cannot be expressed as the sum of distinct primes of the form $6 n - 1$
- 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
- 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: Integers whose Number of Representations as Sum of Two Primes is Maximum
- Previous ... Next: Highly Composite Number
- Previous ... Next: Superabundant Number
- Previous ... Next: Numbers of Zeroes that Factorial does not end with
- Previous ... Next: Integers not Expressible as Sum of Distinct Primes of form 6n-1
- 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 angle 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): $60$
Categories:
- Unitary Perfect Numbers/Examples
- Special Highly Composite Numbers/Examples
- Pyramidal Numbers/Examples
- Highly Abundant Numbers/Examples
- Highly Composite Numbers/Examples
- Superabundant Numbers/Examples
- Integers not Expressible as Sum of Distinct Primes of form 6n-1/Examples
- Semiperfect Numbers/Examples
- Specific Numbers
- 60