Another Prime To Remember

In November of 2024, I created a post titled A Prime To Remember. The prime on that occasion was 27617 and you can read about its properties by following the link. I think it's time to celebrate another prime and that prime is \( \textbf{27947}\). I sometimes struggle to find a single interesting property for the number, on any given day, that is associated with my diurnal age. With 27947 I had no such problem.

First and foremost, it has the property that the sum of its digits, the sum of the squares of its digits and the sum of cubes of its digits are all prime. Thus we have:

• \(2+7+9+4+7 = 29\) 
• \(2^2+7^2+9^2+4^2+7^2 = 199\)
• \(2^3+7^3+9^3+4^3+7^3 = 1487\)

This property affords it membership in OEIS A176179 and there are 322 such numbers in the range up to 40,000. 27947 shares this property with 27617 and so it is included in my blog post A Prime To Remember. 

27947 also has the property that the absolute differences between successive pairs of digits, and also the first and last digits, are all prime. I discuss these sorts of primes in my blog post titled Fun With Primes and Digit Pairs. Thus we have:

• \( | 2 - 7 | = 5\)
• \(| 7 - 9 | = 2\)
• \(| 9 - 4 | = 5\)
• \(| 7 - 2 | = 5\)

This property affords it membership in OEIS A087593. This next property relates to the prime producing quadratic polynomial \( (4n-29)^2 + 58 \). This polynomial generates 28 distinct primes in succession from \(n=1\) to \(n=28\). When \(n=49\), the polynomial produces the prime 27947. This property affords it membership of OEIS A320772. See my blog post Another Prime Generating Polynomial.

Still on the subject of primes, 27947 has the property that it is a balanced prime of order 100 and thus a member of OEIS A363168. A prime \(p\) is in this sequence if the sum of the 100 consecutive primes just less than \(p\), plus \(p\), plus the sum of the 100 consecutive primes just greater than \(p\), divided by 201 equals \(p\). In the case of 27947, we have:$$ \begin{align} p_{3050} &= 27947 \\ p_{2950} &= 26891\\ p_{3150} &= 28933\\ \sum_{n=2950}^{3049} p_n &=2742922 \\ \sum_{n=3051}^{3150} p_n &= 2846478 \\ \text{average } &= \frac{ 2742922+27947 + 2846478}{201} \\ &= 27947 \end{align}$$See my blog post titled Varieties of Balanced Primes.