An Interesting Triple 7 Number

Today I turned $\mathbf{\text{27717}}$ days old and this number has a plethora of interesting properties that deserve a special mention and thus a dedicated post. Here are some of those properties.

• $\mathbf{\text{27717}}$ is a so-called Lucky Cube, meaning it is a number whose cubes contain the digit sequence “888”, here:$${27717}^3=21293088810813$$The numbers that satisfy from 27717 to 40000 are: 27717, 27942, 27973, 28192, 28442, 28484, 28692, 28740, 28942, 29079, 29192, 29354, 29387, 29391, 29418, 29420, 29442, 29491, 29642, 29692, 29942, 29989.
  
  
• $\mathbf{\text{27717}}$ is the lesser of a pair of adjacent composite numbers such that both are only one step away from their home primes. Here: 
• $27717=3\times 9239\to 39239$
• $27718=2\times 13859\to 213859$
  
  
• $\mathbf{\text{27717}}$ is a number such that n + POD(n) and n - POD(n) are both prime (where POD stands for Product Of Digits). Here we have POD = 686:
• $27717+686=28403$ which is a prime number
• $27717-686=27031$ which is a prime number
  
  
• $\mathbf{\text{27717}}$ is an interprime number because it is at equal distance from the previous prime (27701) and the next prime (27733).
  
  
• $\mathbf{\text{27717}}$ is a number whose sum of divisors has prime factors (ignoring multiplicity) that multiply to the factorial 2310 where
  
  $2310=2\times 3\times 5\times 7\times 11$
  
  Here 27717 has a sum of divisors 36960 and
  
  $36960=2^5\times 3\times 5\times 7\times 11$
  
  but also forms a consecutive pair with 27718 because its sum of the divisors is 41580 and
  
  $41580=2^2\times 3^3\times 5\times 7\times 11$
  
  See blog post Primorials and the Sigma Function.
  
  
• $\mathbf{\text{27717}}$ is the TENTH member of an interesting number chain (which is base independent):
  
• $27708=12\times 2309$
• $27709=11\times 2519$
• $27710=10\times 2771$
• $27711=9\times 3079$
• $27712=8\times 3464$
• $27713=7\times 3959$
• $27714=6\times 4619$
• $27715=5\times 5543$
• $27716=4\times 6929$
• $27717=3\times 9239$
• $27718=2\times 13859$
  

See blog post Count Down Number Chains  

 

• $\mathbf{\text{27717}}$ is a cyclic number.
  
  
• $\mathbf{\text{27717}}$ is a xenodrome in base 9 : 42016. See blog post Xenodromes.
  
  
• $\mathbf{\text{27717}}$ is a number that does not reach a palindrome after 2001 cycles of the reverse and add algorithm.
  
  
• $\mathbf{\text{27717}}$ is a D-number meaning it is a number $n>3$ such that n divides $k^{n-2}-k$ for all $1<k<n$ relatively prime to $n$.
  
  
• $\mathbf{\text{27717}}$ can be rendered as a digit equation as follows: $2-{\displaystyle \frac{7}{7}}=1^7$