48

Number
$48$ (forty-eight) is:


 * $2^4 \times 3$


 * The $8$th highly composite number after $1, 2, 4, 6, 12, 24, 36$:
 * $\tau \left({48}\right) = 10$


 * The $16$th highly abundant number after $1, 2, 3, 4, 6, 8, 10, 12, 16, 18, 20, 24, 30, 36, 42$:
 * $\sigma \left({48}\right) = 124$


 * The $8$th superabundant number after $1, 2, 4, 6, 12, 24, 36$:
 * $\dfrac {\sigma \left({48}\right)} {48} = \dfrac {124} {48} = 2 \cdotp 58 \dot 3$


 * The $11$th semiperfect number after $6, 12, 18, 20, 24, 28, 30, 36, 40, 42$:
 * $48 = 8 + 16 + 24$


 * The $16$th Ulam number after $1$, $2$, $3$, $4$, $6$, $8$, $11$, $13$, $16$, $18$, $26$, $28$, $36$, $38$, $47$:
 * $48 = 1 + 47$


 * The $2$nd of the largest known pair of Ulam numbers which differ by $1$:
 * $47 = 11 + 36, \ 48 = 1 + 47$


 * The smallest composite number the product of whose proper divisors form its $4$th power:
 * $1 \times 2 \times 3 \times 4 \times 6 \times 8 \times 12 \times 16 \times 24 = 48^4$


 * The $22$nd after $1, 2, 4, 5, 6, 8, 9, 12, 13, 15, 16, 17, 20, 24, 25, 27, 28, 32, 35, 26, 39$ of the $24$ positive integers which cannot be expressed as the sum of distinct non-pythagorean primes.


 * With $75$, an element of the $1$st quasiamicable pair:
 * $\sigma \left({48}\right) = \sigma \left({75}\right) = 124 = 48 + 75 + 1$


 * The smallest positive integer which can be expressed as the sum of $2$ odd primes in $5$ ways:
 * $48 = 43 + 5 = 41 + 7 = 37 + 11 = 31 + 17 = 29 + 19$


 * The $9$th positive integer $n$ after $5, 11, 17, 23, 29, 30, 36, 42$ such that no factorial of an integer can end with $n$ zeroes.


 * The $22$nd positive integer which is not the sum of $1$ or more distinct squares:
 * $2, 3, 6, 7, 8, 11, 12, 15, 18, 19, 22, 23, 24, 27, 28, 31, 32, 33, 43, 44, 47, 48, \ldots$

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