225 (two hundred
ndtwenty-five) is the natural number following
224 and preceding
226.
In mathematics
225 is the smallest number that is a
polygonal number
In mathematics, a polygonal number is a Integer, number that counts dots arranged in the shape of a regular polygon. These are one type of 2-dimensional figurate numbers.
Polygonal numbers were first studied during the 6th century BC by the Ancien ...
in five different ways. It is a
square number
In mathematics, a square number or perfect square is an integer that is the square (algebra), square of an integer; in other words, it is the multiplication, product of some integer with itself. For example, 9 is a square number, since it equals ...
,
an
octagonal number
In mathematics, an octagonal number is a figurate number. The ''n''th octagonal number ''o'n'' is the number of dots in a pattern of dots consisting of the outlines of regular octagons with sides up to ''n'' dots, when the octagons are overlai ...
, and a
squared triangular number
In number theory, the sum of the first cubes is the square of the th triangular number. That is,
:1^3+2^3+3^3+\cdots+n^3 = \left(1+2+3+\cdots+n\right)^2.
The same equation may be written more compactly using the mathematical notation for summa ...
.
As the square of a
double factorial
In mathematics, the double factorial of a number , denoted by , is the product of all the positive integers up to that have the same Parity (mathematics), parity (odd or even) as . That is,
n!! = \prod_^ (n-2k) = n (n-2) (n-4) \cdots.
Restated ...
, counts the number of
permutation
In mathematics, a permutation of a set can mean one of two different things:
* an arrangement of its members in a sequence or linear order, or
* the act or process of changing the linear order of an ordered set.
An example of the first mean ...
s of six items in which all cycles have even length, or the number of permutations in which all cycles have odd length. And as one of the
Stirling numbers of the first kind, it counts the number of permutations of six items with exactly three cycles.
225 is a
highly composite odd number, meaning that it has more divisors than any smaller odd numbers. After 1 and 9, 225 is the third smallest number ''n'' for which , where ''σ'' is the
sum of divisors function and ''φ'' is
Euler's totient function
In number theory, Euler's totient function counts the positive integers up to a given integer that are relatively prime to . It is written using the Greek letter phi as \varphi(n) or \phi(n), and may also be called Euler's phi function. In ot ...
. 225 is a
refactorable number.
225 is the smallest
square number
In mathematics, a square number or perfect square is an integer that is the square (algebra), square of an integer; in other words, it is the multiplication, product of some integer with itself. For example, 9 is a square number, since it equals ...
to have one of every digit in some number base (225 is 3201 in base 4)
225 is the smallest odd number with exactly nine divisors.
References
Integers
{{Num-stub