288 (number)

Because its prime factorization

$${\displaystyle 288=2^{5}\cdot 3^{2}}$$

contains only the first two prime numbers 2 and 3, 288 is a 3-smooth number.[1] 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.[2][3] Among the highly abundant numbers, numbers with record-setting sums of divisors, it is one of only 13 such numbers with an odd divisor sum.[4]

Both 288 and 289 = 17² are powerful numbers, numbers in which all exponents of the prime factorization are larger than one.[5][6][7] 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.[4][8] 288 and 289 form only the second consecutive pair of powerful numbers after 8 and 9.[5][6][7]

288 is a superfactorial, a product of consecutive factorials, since[5][9][10]

$${\displaystyle 288=1!\cdot 2!\cdot 3!\cdot 4!=1^{4}\cdot 2^{3}\cdot 3^{2}\cdot 4^{1}.}$$

Coincidentally, as well as being a product of descending powers, 288 is a sum of ascending powers:[11]

$${\displaystyle 288=1^{1}+2^{2}+3^{3}+4^{4}.}$$

288 appears prominently in Stirling's approximation for the factorial, as the denominator of the second term of the Stirling series[12]

$${\displaystyle n!\sim {\sqrt {2\pi n}}\left({\frac {n}{e}}\right)^{n}\left(1+{\frac {1}{12n}}+{\frac {1}{288n^{2}}}-{\frac {139}{51840n^{3}}}-{\frac {571}{2488320n^{4}}}+\cdots \right).}$$

288 is connected to the figurate numbers in multiple ways. It is a pentagonal pyramidal number[13][14] and a dodecagonal number.[14][15] Additionally, it is the index, in the sequence of triangular numbers, of the fifth square triangular number:[14][16]

$${\displaystyle 41616={\frac {288\cdot 289}{2}}=204^{2}.}$$

There are 288 different ways of completely filling in a ${\displaystyle 4\times 4}$ sudoku puzzle grid.[17][18] For square grids whose side length is the square of a prime number, such as 4 or 9, a completed sudoku puzzle is the same thing as a "pluperfect Latin square", an ${\displaystyle n\times n}$ array in which every dissection into ${\displaystyle n}$ rectangles of equal width and height to each other has one copy of each digit in each rectangle. Therefore, there are also 288 pluperfect Latin squares of order 4.[19] There are 288 different ${\displaystyle 2\times 2}$ invertible matrices modulo six,[20] and 288 different ways of placing two chess queens on a ${\displaystyle 6\times 6}$ board with toroidal boundary conditions so that they do not attack each other.[21] There are 288 independent sets in a 5-dimensional hypercube, up to symmetries of the hypercube.[22]