153 (number) - Mathematical Properties

Mathematical Properties

As a triangular number, 153 is the sum of the first 17 integers, and is also the sum of the first five positive factorials:.

The number 153 is also a hexagonal number, and a truncated triangle number, meaning that 1, 15, and 153 are all triangle numbers.

The distinct prime factors of 153 add up to 20, and so do the ones of 154, hence the two form a Ruth-Aaron pair.

Since, it is a 3-narcissistic number, and it is also the smallest three-digit number which can be expressed as the sum of cubes of its digits. Only five other numbers can be expressed as the sum of the cubes of their digits: 0, 1, 370, 371 and 407. It is also a Friedman number, since 153 = 3 × 51, and a Harshad number in base 10, being divisible by the sum of its own digits.

The Biggs–Smith graph is a symmetric graph with 153 edges, all equivalent.

Another interesting feature of the number 153 is that it is the limit of the following algorithm:

  1. Take a random positive integer, divisible by three.
  2. Split that number into its base 10 digits.
  3. Take the sum of their cubes.
  4. Go back to the second step.

An example, starting with the number 84:

\begin{align} 8^3 + 4^3 &=& 512 + 64 &=& 576\\ 5^3 + 7^3 + 6^3 &=& 125 + 343 + 216 &=& 684\\ 6^3 + 8^3 + 4^3 &=& 216 + 512 + 64 &=& 792\\ 7^3 + 9^3 + 2^3 &=& 343 + 729 + 8 &=& 1080\\ 1^3 + 0^3 + 8^3 + 0^3 &=& 1 + 0 + 512 + 0 &=& 513\\ 5^3 + 1^3 + 3^3 &=& 125 + 1 + 27 &=& 153\\ 1^3 + 5^3 + 3^3 &=& 1 + 125 + 27 &=& 153 \end{align}

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