# 48

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## Number

$48$ (**forty-eight**) is:

- $2^4 \times 3$

- The $1$st 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 $1$st positive integer which can be expressed as the sum of $2$ odd primes in $5$ ways:
- $48 = 43 + 5 = 41 + 7 = 37 + 11 = 31 + 17 = 29 + 19$

- With $75$, an element of the $1$st quasiamicable pair:
- $\map \sigma {48} = \map \sigma {75} = 124 = 48 + 75 + 1$

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

- The $8$th highly composite number after $1$, $2$, $4$, $6$, $12$, $24$, $36$:
- $\map \tau {48} = 10$

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

- 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 $11$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 $16$th highly abundant number after $1$, $2$, $3$, $4$, $6$, $8$, $10$, $12$, $16$, $18$, $20$, $24$, $30$, $36$, $42$:
- $\map \sigma {48} = 124$

- The $18$th of $35$ integers less than $91$ to which $91$ itself is a Fermat pseudoprime:
- $3$, $4$, $9$, $10$, $12$, $16$, $17$, $22$, $23$, $25$, $27$, $29$, $30$, $36$, $38$, $40$, $43$, $48$, $\ldots$

- 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

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

## Also see

*Previous ... Next*: Semiperfect Number*Previous ... Next*: Numbers of Zeroes that Factorial does not end with*Previous ... Next*: Highly Abundant Number

## Sources

- 1986: David Wells:
*Curious and Interesting Numbers*... (previous) ... (next): $48$ - 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): $48$