225 (number)

225 (two hundred [and] twenty-five) is the natural number following 224 and preceding 226.

225 is the smallest number that is a polygonal number in five different ways.^[1] It is a square number (225 = 15²),^[2] an octagonal number,^[3] and a squared triangular number (225 = (1 + 2 + 3 + 4 + 5)² = 1³ + 2³ + 3³ + 4³ + 5³) .^[4]

As the square of a double factorial, 225 = 5!!² counts the number of permutations of six items in which all cycles have even length, or the number of permutations in which all cycles have odd length.^[5] And as one of the Stirling numbers of the first kind, it counts the number of permutations of six items with exactly three cycles.^[6]

225 is a highly composite odd number, meaning that it has more divisors than any smaller odd numbers.^[7] After 1 and 9, 225 is the third smallest number n for which σ(φ(n)) = φ(σ(n)), where σ is the sum of divisors function and φ is Euler's totient function.^[8] 225 is a refactorable number.^[9]

225 is the smallest square number to have one of every digit in some number base (225 is 3201 in base 4) ^[10]

225 is the first odd number with exactly 9 divisors.^[11]

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