Sphenic Numbers Revisited

After this post, I discovered that I'd already made an earlier post about sphenic numbers. No matter but it alerted me to the fact that I've made so many posts to this mathematics blog that I'm losing track of what I've posted.

Today I turned 25474 days old. This number factors to 2 * 47 * 271. Yesterday's number, 25473, factors to 3 * 7 * 1213. Both are sphenic numbers, described by Numbers Aplenty as follows:

A number \(n\) is called sphenic if it is the product of 3 distinct primes. For example, 370 is a sphenic number because it is the product of the 3 primes 2, 5 and 37. Sphenic numbers are quite common: up to \(10^8\) there are 20710806 sphenic numbers (that's about 20%). 

The sum of the reciprocals of the sphenic numbers diverges, while the sum of the reciprocal of their squares converges to \(0.003696244...\), which can be expressed as: $$ \frac{(P(2)^3-3 \, P(2) \, P(4)+2 \,P(6))}{6}\\ \text { where }P(s)=\sum_{p\mathrm{\ prime}}\frac{1}{p^s}$$ is the so-called prime Zeta function. 

The first sphenic numbers are 30, 42, 66, 70, 78, 102, 105, 110, 114, 130, 138, 154, 165, 170, 174, 182, 186, 190, 195, 222, 230, 231, 238, 246, 255, 258, 266, 273, 282, 285, 286, 290, 310

Wikipedia adds that:

All sphenic numbers have exactly eight divisors. If we express the sphenic number as \( n = p \cdot q \cdot r\) where \(p\), \(q\), and \(r\) are distinct primes, then the set of divisors of \(n\) will be \({1, p, q, r, pq, pr, qr, n}\). The converse does not hold. For example, 24 is not a sphenic number, but it has exactly eight divisors. 

All sphenic numbers are by definition squarefree, because the prime factors must be distinct. 

The first case of two consecutive sphenic integers is 230 = 2×5×23 and 231 = 3×7×11. The first case of three is 1309 = 7×11×17, 1310 = 2×5×131, and 1311 = 3×19×23. There is no case of more than three, because every fourth consecutive positive integer is divisible by 4 = 2×2 and therefore not squarefree. 

The numbers 2013 (3×11×61), 2014 (2×19×53), and 2015 (5×13×31) are all sphenic. It's interesting that these very recent calendar years formed a sphenic triplet, although I didn't know it at the time I was living through them. The next three consecutive sphenic years will be 2665 (5×13×41), 2666 (2×31×43) and 2667 (3×7×127) (see OEIS A248202 for a list of the central number of such triples). 

Sphenic Brick

In terms of geometry, each sphenic number can be considered to represent the volume of a unique and "primitive" rectangular prism (sometimes called a sphenic brick) whose dimensions are given by its three prime factors. I'm using primitive here in the same sense as "primitive Pythagorean triad" such as 3, 4 and 5 (as opposed to 6, 8 and 10). By its definition however, a sphenic number can never represent the volume of a cube or a rectangular prism with a square cross-section.

Each sphenic number \( n = p \cdot q \cdot r\) can be associated with another number, namely the surface area of the rectangular prism  \( 2 \, (p \cdot q + p \cdot r+q \cdot r) \). For example, the sphenic number  \( 7429 = 17 \cdot 19 \cdot 23\) can be viewed as a rectangular prism with an associated surface area of \(2302\) square units. The ratio between area and volume can then be explored. The table below shows the values of such ratios for sphenic numbers between 25400 and 25500:

It is possible for the volume and surface area to be equal. In a range of numbers between 1 and 1000, the only such dimensions that produce this are:

• 3 x 7 x 42   --> 882 
• 3 x 8 x 24   --> 576
• 3 x 9 x 18   --> 486
• 3 x 10 x 15 --> 450
• 4 x 5 x 20   --> 400
• 4 x 6 x 12   --> 288

Whether these are the only values with this property I don't know but none of the above numbers (288, 400, 450, 486, 576 and 882) are sphenic so it's likely that there are no sphenic numbers with this property.

To determine the sphenic numbers within a given range, this SageMath code (link to SageMathCell server) can be used or the box below (sometimes temperamental) can be experimented with: