Apéry's Constant

Today I turned 25355 days old and this number turns up in the Engel expansion of \( \zeta(3) \). Firstly however, let's remind ourselves that \( \zeta \) is the Riemann zeta function and can be expressed as:$$ \zeta(3)=\sum_1^{\infty} \frac{1}{n^3} $$ $$ \text{or  } \zeta(3)=\lim_{n \rightarrow \infty} \left( \frac{1}{1^3}+\frac{1}{2^3}+ \cdots + \frac{1}{n^3} \right) $$This works out to around 1.202056903159594285399738161511449990764986292 and is known as Apèry's constant. I've written about this constant before in a post titled The Basel Problem and Beyond on May 7th 2017. According to Wikipedia:

This constant arises naturally in a number of physical problems, including in the second-and third-order terms of the electron's gyromagnetic ratio using quantum electrodynamics. It also arises in the analysis of random minimum spanning trees and in conjunction with the gamma function when solving certain integrals involving exponential functions in a quotient which appear occasionally in physics, for instance when evaluating the two-dimensional case of the Debye model and the Stefan–Boltzmann law.

This doesn't mean much to me but there you have it. As for the Engel expansion, I made a blog post about it in 2016. The expansion consists of the terms that go to make up the denominators of the fractions that when added will approximate \( \zeta(3) \). OEIS A053980 lists the first few terms as: 1, 5, 98, 127, 923, 5474, 16490, 25355

A053980

Engel expansion of zeta(3) = 1.20206...

This means that \( \zeta(3) \) can be approximated as:$$ \zeta(3) \approx \frac{1}{1}+\frac{1}{1 \times 5} +\frac{1}{1 \times 5 \times 98}+\frac{1}{1 \times 5 \times 98 \times 127} +\frac{1}{1 \times 5 \times 98 \times 127} + \cdots $$Of course, once the zeta function is touched upon, one can find oneself in very deep water very quickly so I'm not going to say too much more except to include a screenshot (Figure 1) from Wolfram MathWorld showing the Engel expansions for some of the other constants:

Figure 1

The SAGE code for generating these sequences is fairly straightforward. Figure 2 shows what's involved for Apèry's constant:

Figure 2: permalink

Of course, simply replacing u in the above code by say \(e\) or \(\pi \) will produce the associated Engel expansion.

on May 12th 2021

Sphenic Numbers

Today I turned 25285 days old and, as I discovered in Numbers Aplenty, 25285 is a sphenic number. The definition given on that site is:

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 20,710,806 sphenic numbers. 

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  \((P(2)^3-3\,P(2)\,P(4)+2\,P(6))/6\), 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

The sum of the reciprocals of the squares of the sphenic numbers does indeed approach 0.003696244 as can be seen by taking the numbers from 30 to 310 and applying the following SAGE code:

INPUT: 

sphenic=[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]
sum=0
for n in sphenic:
    sum+=1/n^2
print(sum.n())

OUTPUT: 

0.00320320889263633 

The following is a modification of the Wikipedia entry for sphenic numbers:

In number theory, a sphenic number is a natural number|positive integer that is the product of three distinct prime numbers. Thus a sphenic number is a product ''pqr'' where ''p'', ''q'', and ''r'' are three distinct prime numbers. This definition is more stringent than simply requiring the integer to have exactly three prime factors. For instance, \(60 = 2^2 × 3 × 5 \) has exactly 3 prime factors, but is not sphenic. 

The smallest sphenic number is 30 = 2 × 3 × 5, the product of the smallest three primes. The largest known sphenic number is
$$(2^{77232917}− 1) \cdot (2^{74,207,281} − 1) \cdot (2^{57,885,161} − 1)$$It is the product of the three largest known primes. 

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 Möbius function of any sphenic number is -1. 

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 square-free. 

The numbers 2013 (3×11×61), 2014 (2×19×53), and 2015 (5×13×31) are all sphenic. The next three consecutive sphenic years will be 2665 (5×13×41), 2666 (2×31×43) and 2667 (3×7×127) (see OEIS A165936).

The OEIS sequence A007304 (sphenic numbers, products of 3 distinct primes) mentions that a sphenic brick is a rectangular parallelopiped whose sides are components of a sphenic number, namely whose sides are three distinct primes. For example, the distinct prime triple (3,5,7) produces a 3 x 5 x 7 unit brick which has volume 105 cubic units. 

From my early days of investigating numbers, I had considered triprimes (number that factor into three, not necessarily distinct, primes) as having unique representations as rectangular prisms. I was interested in the relationship between a prisms volume and its surface area, coining the term "cubicity". Here is a snapshot of a February 2014 Instagram post: