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From Wikipedia, the free encyclopedia

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]

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In other fields

References

  1. ^ 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.
  2. ^ Sloane, N. J. A. (ed.). "Sequence A000290 (The squares)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
  3. ^ Sloane, N. J. A. (ed.). "Sequence A000567 (Octagonal numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
  4. ^ 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.
  5. ^ Sloane, N. J. A. (ed.). "Sequence A001818 (Squares of double factorials)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
  6. ^ 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.
  7. ^ Sloane, N. J. A. (ed.). "Sequence A053624 (Highly composite odd numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
  8. ^ 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.
  9. ^ "Sloane's A033950 : Refactorable numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. 2016-04-18. Retrieved 2016-04-18.
  10. ^ 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.
This page was last edited on 30 August 2019, at 07:13
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