Some Interesting Properties of 39

My daughter-in-law turned 39 yesterday and so I was prompted to investigate some of its mathematical properties. One of its properties is its membership in OEIS A055233:

A055233: composite numbers equal to the sum of the primes from their smallest prime factor to their largest prime factor.

The only members of this sequence in the range up to 40000 are 10, 39, 155 and 371. All are semiprimes and factorise as follows:

• \(10 = 2 \times 5 \text{ with } 2 + 3 + 5 = 10 \)
• \(39 = 3 \times 13 \text{ with } 3 + 5 + 7 + 11+13 = 39\)
• \(155 = 5 \times 31 \text{ with } 5 + 7 + \ldots + 29 + 31=155\)
• \(371 = 7 \times 53 \text{ with } 7 + 11 + \ldots + 47 + 53=371\)

Because they are semiprimes they are thus equal to the product of their smallest and largest prime factors. However, this is not the case for the next member of the sequence: 2935561623745. The reason is that it is not a semiprime.

• \(2935561623745= 5 \times 19 \times 53 \times 61 \times 9557887\)

The next member of the sequence 454539357304421 is a semiprime and thus follows the pattern of the first four members of the sequence:

• \(454539357304421 = 3536123 \times 128541727\)

So we see that 39 by virtue of its membership in OEIS A055233 is rather special. Of course, it has some other interesting qualities. For example, it can be constructed from the first three powers of 3:$$39=3+3^2+3^3$$Gemini also mentions the following number properties:

Beyond these patterns, 39 is also classified as a \( \textbf{Perrin number}\) and a \( \textbf{Størmer number}\), placing it within specialized mathematical sequences that are far from intuitive. 

The number also has an \( \textbf{aliquot sum}\) of 17, which is a prime number, a unique characteristic that links it to a specific aliquot sequence. 

In the realm of number partitions, 39 is notable as the smallest natural number to have three distinct partitions into three parts that all yield the same product, 1200. These partitions are:

• {25, 8, 6} 

• {24, 10, 5} 

• {20, 15, 4}. 

Lastly, in analytic number theory, the \( \textbf{Mertens function}\) returns a value of 0 when given 39, a property that suggests a form of numerical equilibrium or stability, a concept that finds intriguing parallels in other domains. See blog post Zeroes of the Mertens Function.

39 is also what's termed a \( \textbf{perfect totient number} \) because the sum of its iterated totients equals the number itself. Let's confirm this:$$ \begin{align} \phi(39) &=24 \\ \phi(24) &=8 \\ \phi(8) &=4 \\ \phi(4) &=2 \\ \phi(2) &=1 \end{align} $$The sum of these iterated totients equals 39:$$24 + 8 + 4 + 2 + 1 =39$$The perfect totient numbers are listed in OEIS A082897 (permalink):

3, 9, 15, 27, 39, 81, 111, 183, 243, 255, 327, 363, 471, 729, 2187, 2199, 3063, 4359, 4375, 5571, 6561, 8751, 15723, 19683, 36759, 46791, 59049, 65535, 140103, 177147, 208191, 441027, 531441, 1594323, 4190263, 4782969, 9056583, 14348907, 43046721