Record Runs Involving Home Primes

I just happened to notice that there is a run of six consective numbers from 27597 to 27602 inclusive that are only one step removed from their home primes. The only longer run up to one million that occurs is a run of eight numbers from 45162 to 45169.

Here is a permalink for finding these runs. Currently I'm 27586 days old and so the coming record run is not far off. Here are the factorisations for the numbers and the home primes associated with them.

• \(27597 = 3 \times 9199 \rightarrow 39199\)
  
  
• \(27598 = 2  \times13799 \rightarrow 213799 \)
  
  
• \(27599 = 11 \times 13 \times 193 \rightarrow 1113193 \)
  
  
• \(27600 = 2^4 \times 3 \times 5^2 \times  23 \rightarrow 222235523 \)
  
  
• \(27601 = 7 \times 3943 \rightarrow 73943 \)
  
  
• \(27602 = 2 \times 37 \times 373 \rightarrow 237373 \)

The run of eight numbers is listed below together with factorisations and home primes:

• \(45162 = 2 \times 3^2 \times 13 \times 193 \rightarrow 23313193 \)
  
  
• \(45163 = 19 \times 2377 \rightarrow 192377 \)
  
  
• \(45164 = 2^2 \times 7 \times 1613 \rightarrow 2271613 \)
  
  
• \(45165 = 3 \times 5 \times 3011\rightarrow 353011 \)
  
  
• \(45166 = 2 \times 11 \times 2053 \rightarrow 2112053 \)
  
  
• \(45167 = 31^2 \times 47 \rightarrow 313147\)
  
  
• \(45168 = 2^4 \times 3 \times 941 \rightarrow 22223941 \)
  
  
• \(45169 = 17 \times 2657 \rightarrow 172657 \)
  

All concatenations are applied to the prime factors in order from lowest to highest. Of course, runs formed by numbers that are concatenations of prime factors from highest to lowest are impossible because every second number is even with a smallest factor of 2. Every second concatenated number will thus be even as well.