240

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Number

$240$ (two hundred and forty) is:

$2^4 \times 3 \times 5$

The length of the longest edge of the smallest cuboid whose edges and the diagonals of whose faces are all integers:

The lengths of the edges are $44, 117, 240$

The lengths of the diagonals of the faces are $125, 244, 267$.

The $9$th positive integer after $64$, $96$, $128$, $144$, $160$, $192$, $216$, $224$ with $6$ or more prime factors:

$240 = 2 \times 2 \times 2 \times 2 \times 3 \times 5$

The $12$th highly composite number after $1$, $2$, $4$, $6$, $12$, $24$, $36$, $48$, $60$, $120$, $180$:

$\map {\sigma_0} {240} = 20$

The $12$th superabundant number after $1$, $2$, $4$, $6$, $12$, $24$, $36$, $48$, $60$, $120$, $180$:

$\dfrac {\map {\sigma_1} {240} } {240} = \dfrac {744} {240} = 3 \cdotp 1$

The $29$th highly abundant number after $1$, $2$, $3$, $4$, $6$, $8$, $10$, $12$, $16$, $18$, $20$, $24$, $30$, $36$, $42$, $48$, $60$, $72$, $84$, $90$, $96$, $108$, $120$, $144$, $168$, $180$, $210$, $216$:

$\map {\sigma_1} {240} = 744$

An integer with over $240$ divisors is greater than $1 \, 000 \, 000$.

Arithmetic Functions on $240$

\(\ds \map {\sigma_0} { 240 }\)

\(=\)

\(\ds 20\)

$\sigma_0$ of $240$

\(\ds \map {\sigma_1} { 240 }\)

\(=\)

\(\ds 744\)

$\sigma_1$ of $240$

Also see

• Cuboid with Integer Edges and Face Diagonals
• Number with over 240 Divisors is greater than 1,000,000

• Previous  ... Next: Highly Composite Number
• Previous  ... Next: Superabundant Number

• Previous  ... Next: Highly Abundant Number

• Previous  ... Next: Numbers with 6 or more Prime Factors

Sources

• 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $240$
• 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $240$