# 28

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## Number

$28$ (**twenty-eight**) is:

- $2^2 \times 7$

- The only perfect number which is the sum of equal powers of exactly $2$ positive integers:
- $28 = 1^3 + 3^3$

- The $1$st triangular number which is the sum of $2$ cubes:
- $28 = 1 + 27 = 1^3 + 3^3$

- The $2$nd perfect number after $6$:
- $28 = 1 + 2 + 4 + 7 + 14 = \map {\sigma_1} {28} - 28 = 4 \times 7 = 2^{3 - 1} \paren {2^3 - 1}$

- The $3$rd Ore number after $1$, $6$:
- $\dfrac {28 \times \map {\sigma_0} {28} } {\map {\sigma_1} {28} } = 3$

- The $3$rd Keith number after $14$, $19$:
- $2$, $8$, $10$, $18$, $28$, $\ldots$

- The $3$rd primitive semiperfect number after $6$, $20$:
- $28 = 1 + 2 + 4 + 7 + 14$

- The $3$rd Sierpiński number of the first kind after $2$, $5$:
- $28 = 3^3 + 1$

- The $4$th hexagonal number after $1$, $6$, $15$:
- $28 = 1 + 5 + 9 + 13 = 4 \paren {2 \times 4 - 1}$

- The $5$th term of Göbel's sequence after $1$, $2$, $3$, $5$, $10$:
- $28 = \paren {1 + 1^2 + 2^2 + 3^2 + 5^2 + 10^2} / 5$

- The $6$th semiperfect number after $6$, $12$, $18$, $20$, $24$:
- $28 = 1 + 2 + 4 + 7 + 14$

- The $7$th triangular number after $1$, $3$, $6$, $10$, $15$, $21$:
- $28 = 1 + 2 + 3 + 4 + 5 + 6 + 7 = \dfrac {7 \times \paren {7 + 1} } 2$

- Hence there are $28$ dominoes in a standard set

- The $7$th happy number after $1$, $7$, $10$, $13$, $19$, $23$:
- $28 \to 2^2 + 8^2 = 4 + 64 = 68 \to 6^2 + 8^2 = 36 + 64 = 100 \to 1^2 + 0^2 + 0^2 = 1$

- The $12$th Ulam number after $1$, $2$, $3$, $4$, $6$, $8$, $11$, $13$, $16$, $18$, $26$:
- $28 = 2 + 26$

- The $12$th even number after $2$, $4$, $6$, $8$, $10$, $12$, $14$, $16$, $20$, $22$, $26$ which cannot be expressed as the sum of $2$ composite odd numbers.

- The $15$th positive integer which is not the sum of $1$ or more distinct squares:
- $2$, $3$, $6$, $7$, $8$, $11$, $12$, $15$, $18$, $19$, $22$, $23$, $24$, $27$, $28$, $\ldots$

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

- The $20$th integer $n$ after $0$, $1$, $2$, $3$, $4$, $5$, $6$, $7$, $8$, $9$, $13$, $14$, $15$, $16$, $18$, $19$, $24$, $25$, $27$ such that $2^n$ contains no zero in its decimal representation:
- $2^{28} = 268 \, 435 \, 456$

## Also see

*Previous ... Next*: Ore Number*Previous ... Next*: Perfect Number

*Previous ... Next*: Göbel's Sequence

*Previous ... Next*: Hexagonal Number

*Previous ... Next*: Keith Number

*Previous ... Next*: Happy Number

*Previous ... Next*: Positive Even Integers not Expressible as Sum of 2 Composite Odd Numbers*Previous ... Next*: Ulam Number

*Previous ... Next*: Powers of 2 with no Zero in Decimal Representation*Previous ... Next*: Numbers not Sum of Distinct Squares*Previous ... Next*: Positive Integers Not Expressible as Sum of Distinct Non-Pythagorean Primes

## Historical Note

There are approximately $28$ days in the lunar cycle.

This is probably the origin of the widespread cultural custom of splitting time both into $7$-day chunks called weeks, and also the year into $12$ approximately $28$-day chunks called months.

There are $28$ pounds avoirdupois in one quarter.

## Sources

- 1986: David Wells:
*Curious and Interesting Numbers*... (previous) ... (next): $28$ - 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): $28$

Categories:

- Sierpiński Numbers of the First Kind/Examples
- Ore Numbers/Examples
- Perfect Numbers/Examples
- Göbel's Sequence/Examples
- Hexagonal Numbers/Examples
- Keith Numbers/Examples
- Primitive Semiperfect Numbers/Examples
- Triangular Numbers/Examples
- Happy Numbers/Examples
- Semiperfect Numbers/Examples
- Ulam Numbers/Examples
- Specific Numbers
- 28