From Wikipedia, the free encyclopedia
225 (two hundred [and] twenty-five ) is the natural number following
224 and preceding
226 .
In mathematics
Natural number
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]
References
^
Sloane, N. J. A. (ed.).
"Sequence A063778 (a(n) = the least integer that is polygonal in exactly n ways)" . The
On-Line Encyclopedia of Integer Sequences . OEIS Foundation.
^
Sloane, N. J. A. (ed.).
"Sequence A000290 (The squares)" . The
On-Line Encyclopedia of Integer Sequences . OEIS Foundation.
^
Sloane, N. J. A. (ed.).
"Sequence A000567 (Octagonal numbers)" . The
On-Line Encyclopedia of Integer Sequences . OEIS Foundation.
^
Sloane, N. J. A. (ed.).
"Sequence A000537 (Sum of first n cubes; or n-th triangular number squared)" . The
On-Line Encyclopedia of Integer Sequences . OEIS Foundation.
^
Sloane, N. J. A. (ed.).
"Sequence A001818 (Squares of double factorials)" . The
On-Line Encyclopedia of Integer Sequences . OEIS Foundation.
^
Sloane, N. J. A. (ed.).
"Sequence A000399 (Unsigned Stirling numbers of first kind s(n,3))" . The
On-Line Encyclopedia of Integer Sequences . OEIS Foundation.
^
Sloane, N. J. A. (ed.).
"Sequence A053624 (Highly composite odd numbers)" . The
On-Line Encyclopedia of Integer Sequences . OEIS Foundation.
^
Sloane, N. J. A. (ed.).
"Sequence A033632 (Numbers n such that sigma(phi(n)) = phi(sigma(n)))" . The
On-Line Encyclopedia of Integer Sequences . OEIS Foundation.
^
"Sloane's A033950 : Refactorable numbers" . The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. 2016-04-18. Retrieved 2016-04-18 .
^
Sloane, N. J. A. (ed.).
"Sequence A061845 (Numbers which have one of every digit in some base)" . The
On-Line Encyclopedia of Integer Sequences . OEIS Foundation.
^
Sloane, N. J. A. (ed.).
"Sequence A000005 (the number of divisors of n)" . The
On-Line Encyclopedia of Integer Sequences . OEIS Foundation.
100,000
1,000,000
10,000,000
100,000,000
1,000,000,000