View source for 225 (number)

'''225''' ('''two hundred [and] twenty-five''') is the natural number following [[224 (number)|224]] and preceding [[226 (number)|226]].
== In mathematics ==
{{Infobox number
| number = 225
| prime = no
}}

225  is the smallest number that is a polygonal number in five different ways.<ref>{{Cite OEIS|A063778|name=a(n) = the least integer that is polygonal in exactly n ways}}</ref> It is a [[Square number|square number]] {{nowrap|1=(225 = 15<sup>2</sup>)}},<ref>{{Cite OEIS|A000290|name=The squares}}</ref>
an octagonal number,<ref>{{Cite OEIS|A000567|name=Octagonal numbers}}</ref> and a [[Squared triangular number|squared triangular number]] {{nowrap|1=(225 = (1 + 2 + 3 + 4 + 5)<sup>2</sup> = 1<sup>3</sup> + 2<sup>3</sup> + 3<sup>3</sup> + 4<sup>3</sup> + 5<sup>3</sup>)}} .<ref>{{Cite OEIS|A000537|name=Sum of first n cubes; or n-th triangular number squared}}</ref>

As the square of a [[Double factorial|double factorial]], {{nowrap|1=225 = 5!!<sup>2</sup>}} counts the number of [[Permutation|permutation]]s of six items in which all cycles have even length, or the number of permutations in which all cycles have odd length.<ref>{{Cite OEIS|A001818|name=Squares of double factorials}}</ref> And as one of the [[Stirling numbers of the first kind]], it counts the number of permutations of six items with exactly three cycles.<ref>{{Cite OEIS|A000399|name=Unsigned Stirling numbers of first kind s(n,3)}}</ref>

225 is a [[Highly composite number|highly composite odd number]], meaning that it has more divisors than any smaller odd numbers.<ref>{{Cite OEIS|A053624|name=Highly composite odd numbers}}</ref> After 1 and 9, 225 is the third smallest number ''n'' for which {{nowrap|1=''σ''(''φ''(''n'')) = ''φ''(''σ''(''n''))}}, where ''σ'' is the [[Divisor function|sum of divisors function]] and ''φ'' is [[Euler's totient function]].<ref>{{Cite OEIS|1=A033632|2=Numbers n such that sigma(phi(n)) = phi(sigma(n))}}</ref> 225 is a [[Refactorable number|refactorable number]].<ref>{{Cite web|url=https://oeis.org/A033950|title=Sloane's A033950 : Refactorable numbers|date=2016-04-18|website=The On-Line Encyclopedia of Integer Sequences|publisher=OEIS Foundation|access-date=2016-04-18}}</ref>

225 is the smallest [[Square number|square number]] to have one of every digit in some number base (225 is 3201 in base 4) <ref>{{Cite OEIS|A061845|name=Numbers which have one of every digit in some base}}</ref>

225 is the first odd number with exactly 9 divisors.<ref>{{Cite OEIS|A000005|name=the number of divisors of n}}</ref>

==References==
{{reflist|30em}}


[[Category:Integers]]


{{Sourceattribution|225 (number)}}