An Interesting Triple 7 Number

Today I turned \( \textbf{27717} \) 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.

• \( \textbf{27717} \) 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.
  
  
• \( \textbf{27717} \) is the lesser of a pair of adjacent composite numbers such that both are only one step away from their home primes. Here: 
• \(27717 = 3 \times 9239 \rightarrow 39239\)
• \(27718 = 2 \times 13859 \rightarrow 213859\)
  
  
• \( \textbf{27717} \) 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
  
  
• \( \textbf{27717} \) is an interprime number because it is at equal distance from the previous prime (27701) and the next prime (27733).
  
  
• \( \textbf{27717} \) 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
  
  \(41580= 2^2 \times 3^3 \times 5  \times 7 \times11\)
  
  See blog post Primorials and the Sigma Function.
  
  
• \( \textbf{27717} \) 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  

 

• \( \textbf{27717} \) is a cyclic number.
  
  
• \( \textbf{27717} \) is a xenodrome in base 9 : 42016. See blog post Xenodromes.
  
  
• \( \textbf{27717} \) is a number that does not reach a palindrome after 2001 cycles of the reverse and add algorithm.
  
  
• \( \textbf{27717} \) 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 \(n\).
  
  
• \( \textbf{27717} \) can be rendered as a digit equation as follows: \(2 - \dfrac{7}{7} = 1 ^ 7\)