# 120

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## Number

$120$ (one hundred and twenty) is:

$2^3 \times 3 \times 5$

The $1$st triperfect number:
$\map {\sigma_1} {120} = 360 = 3 \times 120$

The $2$nd triangular number after $6$ which can be expressed as the product of $3$ consecutive integers:
$120 = T_{15} = 4 \times 5 \times 6$

The $2$nd after $24$ of the $3$ integers which can be expressed as the product of both $3$ and $4$ consecutive integers:
$120 = 4 \times 5 \times 6 = 2 \times 3 \times 4 \times 5$

The $3$rd after $1$, $10$ of the $5$ tetrahedral numbers which are also triangular.

The $4$th and final element of the Fermat set after $1$, $3$, $8$

The $5$th factorial after $1$, $2$, $6$, $24$:
$120 = 5! = 5 \times 4 \times 3 \times 2 \times 1$

The $6$th untouchable number after $2$, $5$, $52$, $88$, $96$.

The smallest positive integer greater than $1$ to appear $6$ times in Pascal's Triangle.

The $8$th hexagonal number after $1$, $6$, $15$, $28$, $45$, $66$, $91$:
$120 = 1 + 5 + 9 + 13 + 17 + 21 + 25 + 29 = 8 \paren {2 \times 8 - 1}$

The $8$th tetrahedral number, after $1$, $4$, $10$, $20$, $35$, $56$, $84$:
$120 = 1 + 3 + 6 + 10 + 15 + 21 + 28 + 36 = \dfrac {8 \paren {8 + 1} \paren {8 + 2} } 6$

The $10$th highly composite number after $1$, $2$, $4$, $6$, $12$, $24$, $36$, $48$, $60$:
$\map {\sigma_0} {120} = 16$

The $10$th superabundant number after $1$, $2$, $4$, $6$, $12$, $24$, $36$, $48$, $60$:
$\dfrac {\map {\sigma_1} {120} } {120} = \dfrac {360} {120} = 3$

The smallest positive integer which can be expressed as the sum of $2$ odd primes in $12$ ways.

The $15$th triangular number after $1$, $3$, $6$, $10$, $15$, $21$, $28$, $36$, $45$, $55$, $66$, $78$, $91$, $105$:
$120 = \ds \sum_{k \mathop = 1}^{15} k = \dfrac {15 \times \paren {15 + 1} } 2$

There are $17$ positive integers which have an Euler $\phi$ value $120$.

The $23$rd highly abundant number after $1$, $2$, $3$, $4$, $6$, $8$, $10$, $12$, $16$, $18$, $20$, $24$, $30$, $36$, $42$, $48$, $60$, $72$, $84$, $90$, $96$, $108$:
$\map {\sigma_1} {120} = 360$

### Arithmetic Functions on $120$

 $\ds \map {\sigma_0} { 120 }$ $=$ $\ds 16$ $\sigma_0$ of $120$ $\ds \map {\sigma_1} { 120 }$ $=$ $\ds 360$ $\sigma_1$ of $120$

## Historical Note

The number $120$ was at one time sometimes referred to in England as a long hundred.

Hence the use of the term short hundred for the number $100$.

Both terms are now obsolete in England, although the term great hundred for $120$ is still used in Germany and Scandinavia.