288 (number)

In mathematics

Factorization properties

Because its prime factorization 288 = 2^5\cdot 3^2 contains only the first two prime numbers 2 and 3, 288 is a 3-smooth number. This factorization also makes it a highly powerful number, a number with a record-setting value of the product of the exponents in its factorization.{{cite journal

Both 288 and are powerful numbers, numbers in which all exponents of the prime factorization are larger than one. This property is closely connected to being highly abundant with an odd divisor sum: all sufficiently large highly abundant numbers have an odd prime factor with exponent one, causing their divisor sum to be even.{{cite journal

Factorial properties

288 is a superfactorial, a product of consecutive factorials, since{{cite book

288 appears prominently in Stirling's approximation for the factorial, as the denominator of the second term of the Stirling series n! \sim \sqrt{2\pi n}\left(\frac{n}{e}\right)^n \left(1 +\frac{1}{12n}+\frac{1}{288n^2} - \frac{139}{51840n^3} -\frac{571}{2488320n^4}+ \cdots \right).

Figurate properties

288 is connected to the figurate numbers in multiple ways. It is a pentagonal pyramidal number and a dodecagonal number.{{cite book

Enumerative properties

There are 288 different ways of completely filling in a 4\times 4 sudoku puzzle grid.{{cite journal

In other areas

In early 20th-century molecular biology, some mysticism surrounded the use of 288 to count protein structures, largely based on the fact that it is a smooth number.{{cite journal | doi-access = free

A common mathematical pun involves the fact that and that 144 is named as a gross: "Q: Why should the number 288 never be mentioned? A: it is two gross."{{cite book

References

References

1. {{cite OEIS. A003586. 3-smooth numbers
2. {{cite OEIS. A005934. Highly powerful numbers
3. {{cite OEIS. A128700. Highly abundant numbers with an odd divisor sum
4. A060355. Numbers n such that n and n+1 are a pair of consecutive powerful numbers
5. {{cite OEIS. A000178. Superfactorials
6. {{cite OEIS. A001923
7. {{cite OEIS. A001164. Stirling's formula: denominators of asymptotic series for Gamma function
8. {{cite OEIS. A002411. Pentagonal pyramidal numbers
9. {{cite OEIS. A051624. 12-gonal (or dodecagonal) numbers
10. {{cite OEIS. A001108. a(n)-th triangular number is a square
11. {{cite OEIS. A107739. Number of (completed) sudokus (or Sudokus) of size n^2 X n^2
12. {{cite OEIS. A108395. Number of pluperfect Latin squares of order n
13. {{cite OEIS. A000252. Number of invertible 2 X 2 matrices mod n
14. {{cite OEIS. A172517. Number of ways to place 2 nonattacking queens on an n X n toroidal board
15. {{cite OEIS. A060631. Number of independent sets in an n-dimensional hypercube modulo symmetries of the hypercube