Five Five Numbers

The number associated with my diurnal age today, \( \textbf{27905} \), is one of those numbers that it is difficult to find anything of interest about. However, as usual, a little investigation turned up something special about it. It is what I call a five five number meaning it meets the following criteria:

• it is a composite and squarefree number
• its digits contain a single 5
• each prime factor contains at least one 5 for a total of three 5's
• its arithmetical digital root is 5
  
  

In the range up to 40000, there are only six numbers that satisfy these criteria and they are 12785, 27635, 27815, 27905, 28265 and 32765. Here are the details (permalink):

 number   factors    root

  12785    5 * 2557   5

  27635    5 * 5527   5

  27815    5 * 5563   5

  27905    5 * 5581   5

  28265    5 * 5653   5

  32765    5 * 6553   5

You could extend this idea to digits other than \( \textbf{5} \). The digit \( \textbf{3} \) produces too many suitable numbers but what about the digit \( \textbf{4} \) where we require this of the number:

• it is a composite and squarefree number
• its digits contain a single 4
• each prime factor contains at least one 4 for a total of two 4's
• its arithmetical digital root is 4
  
  

In the range up to 40000, only one number satisfies and that is 19147 = 41 x 467 with a digital root of 4 (permalink). Nothing for the digit \( \textbf{6} \) up to one million. For the digit \( \textbf{7} \) there is only one number up to one million and that is 544579 = 7 x 77797 with a digital root of 7 - we require that its prime factors contain a total of five 7's (permalink). For the digits \( \textbf{8} \) and \( \textbf{9} \), no numbers qualify up to one million.

So it turns out that 27905 is not so uninteresting after all. Additionally, it is a \( \textbf{Proth} \) number, since it is equal to \(109 \times 2^8 + 1\) and \(109 < 28 \). I've written about these before in a post titled Proth Numbers.