Persistent Tau Numbers

I got to thinking about tau numbers, those numbers that are divisible by their number of divisors, because it's relevant to my divisors sequence which arises when these rules are applied recursively to a positive integer. The rules for a number \(n\) are:

Let \(d\) be the number of divisors of \(n\)

• if \(d \,| \,n \) then \( n \rightarrow d \, | \,n \)
• if \( d \, \nmid \, n\) then \(n \rightarrow d \times n\)
• if \(d\) is prime then:
• if \(d\) is a \(4k+1 \) prime then \( d \rightarrow 2 \times d +1\)
• if \(d\) is a \(4k+3 \) prime then \( d \rightarrow \dfrac {d -1}{2}\)

I'll get back to that sequence later but what interested me were those tau numbers that remain tau numbers after being divided by their number of digits. I got Gemini to investigate and the results revealed that there are 160 such numbers in the range from 1 to 40000 (permalink) and they are:

1, 2, 8, 12, 80, 96, 108, 240, 384, 640, 720, 864, 896, 1200, 1408, 1440, 1664, 1920, 2016, 2176, 2240, 2432, 2560, 2688, 2944, 3168, 3456, 3600, 3712, 3744, 3968, 4032, 4224, 4736, 4896, 4992, 5040, 5248, 5472, 5504, 6016, 6528, 6624, 6784, 7296, 7552, 7808, 7920, 8100, 8352, 8576, 8832, 8928, 9088, 9344, 9360, 10112, 10624, 10656, 10800, 11136, 11392, 11808, 11904, 12240, 12384, 12416, 12928, 13184, 13536, 13680, 13696, 13952, 14208, 14464, 15264, 15744, 16256, 16512, 16560, 16768, 16992, 17536, 17568, 17792, 17920, 18048, 19072, 19296, 19328, 20096, 20160, 20352, 20448, 20864, 20880, 21024, 21376, 22144, 22320, 22656, 22752, 22912, 23040, 23168, 23424, 23904, 24300, 24448, 24704, 25216, 25472, 25632, 25728, 26244, 26640, 27008, 27264, 27936, 28032, 28224, 28544, 29056, 29088, 29312, 29520, 29664, 29824, 30000, 30336, 30592, 30720, 30816, 30848, 30960, 31392, 31872, 32128, 32544, 32896, 33664, 33840, 34176, 34432, 34560, 34688, 35456, 35968, 36224, 36576, 37248, 37504, 37728, 38160, 38400, 38784, 39296, 39456, 39552, 39808

Most survive only one division but some go further and these are (in the given range):

• \(80 \rightarrow 8 \rightarrow 2 \rightarrow 1 \)
• \(96 \rightarrow 8 \rightarrow 2 \rightarrow 1 \)
• \(240 \rightarrow 12 \rightarrow 2 \rightarrow 1 \)
• \(2240 \rightarrow 80 \rightarrow 8 \rightarrow 2 \rightarrow 1 \)
• \(3456 \rightarrow 108 \rightarrow 9 \rightarrow 3 \)
• \(3600 \rightarrow 80 \rightarrow 8 \rightarrow 2 \rightarrow 1 \)
• \(4032 \rightarrow 96 \rightarrow 8 \rightarrow 2 \rightarrow 1 \)
• \(23040 \rightarrow 384 \rightarrow 24 \rightarrow 3 \)
• \(30720 \rightarrow 640 \rightarrow 40 \rightarrow 5 \)
• \(38400 \rightarrow 640 \rightarrow 40 \rightarrow 5 \)

I discussed the divisor sequence in my post titled Number's Divisors to Sequence Algorithm. Applied to 23040 (one of the numbers listed in the bullet points above), the sequence is very short indeed: 23040, 384, 24, 3, 1 Once the sequence reaches 3 then, because it is a \(4k+3\) prime, 3 becomes 1 and 1 has one divisor and so 1 times 1 or 1 divided by 1 is still 1.

Only 5.257% of the numbers up to 100,000 are tau numbers, so the divisors to sequence algorithm is strongly weighted toward multiplication. A composite number is about 19 times more likely to get multiplied than it is to get divided. The primes strike a balance because of the \(4k+1\) versus \(4k+3\) rule. The rule is actually not needed as primes always have two divisors. I introduced the prime rule because in the two factors to sequence algorithms (using prime factors with and without multiplicity), the primes have no prime factors.

What if instead of multiplying by the number of divisors, we simply added 1? It would be interesting to see what sort of sequence develops if we do this.

Harmonic Numbers

On January 1st 2024, I made a post titled Unitary Harmonic Numbers which are defined as numbers whose unitary divisors have a harmonic mean that is an integer. Oddly, I have never made a post simply about harmonic numbers defined as numbers whose divisors have a harmonic mean that is an integer. Like unitary harmonic numbers, they are quite rare. The number associated with my diurnal age today (27846) is one such harmonic number. These numbers make up OEIS A001599 and the initial members up to one million are (perfect numbers are shown in red):

1, 6, 28, 140, 270, 496, 672, 1638, 2970, 6200, 8128, 8190, 18600, 18620, 27846, 30240, 32760, 55860, 105664, 117800, 167400, 173600, 237510, 242060, 332640, 360360, 539400, 695520, 726180, 753480, 950976

Here is what Numbers Aplenty had to say about them:

A number \(n\)  is called harmonic divisor number if the harmonic mean of its divisors is an integer. This is equivalent to saying that the average of the divisors of \(n\) divides \(n\): $$ \frac{n}{\sigma(n)/ \tau(n)} = \frac{n \times \tau(n)}{\sigma(n)} \text{ is an integer}$$Harmonic divisor numbers are also called harmonic numbers, for brevity, or Ore numbers, after O.Ore who studied them. He proved that all the perfect numbers are also harmonic and conjectured that 1 is the only odd harmonic number. This conjecture has been verified by G.L.Cohen et al. for \(n<10^{24}\) and if true, it will imply that no odd perfect numbers exist. Jaycob Coleman has observed that all the Ore numbers up to \(10^{14}\) are also practical numbers and conjectured this holds in general. T. Goto and K. Okeya have computed a list of the 937 harmonic numbers up to \(10^{14}\).

In the case of the number (27846) associated with my diurnal age, we have:$$ \begin{align} \tau(27846) &= 48 \\ \sigma(27846) &= 78624\\ \frac{27846 \times 48}{78624} &= 17 \end{align}$$

Strongly Refactorable Numbers

On November 21st 2022, I posted about Tau Numbers and wrote:

A refactorable number or tau number is an integer \(n\) that is divisible by the count of its divisors, or to put it algebraically, \(n\) is such that \( \tau(n) | n \). The first few refactorable numbers are listed in OEIS A033950 as:

1, 2, 8, 9, 12, 18, 24, 36, 40, 56, 60, 72, 80, 84, 88, 96, 104, 108, 128, 132, 136, 152, 156, 180, 184, 204, 225, 228, 232, 240, 248, 252, 276, 288, 296, ...

For example, 18 has 6 divisors (1 and 18, 2 and 9, 3 and 6) and is divisible by 6. There are infinitely many refactorable numbers. Source.

So a refactorable number is the same as a tau number and these are relatively common. Up to 100,000, there are 5257 such numbers representing 5.257% of the range. However, they have a natural density of zero. So what is a strongly refactorable number?

I discovered what characterised these numbers thanks to a property of the number associated with my diurnal age today, 27360, that also corresponds to the 29th of February 2024. It happens to be a member of OEIS A141586:

A141586

Strongly refactorable numbers: numbers \(n\) such that if \(n\) is divisible by \(d\), it is divisible by the number of divisors of \(d\).

Such numbers are few and far between. The number previous to 27360 was 25440. The 41 members of the sequence up to 40,000 are (permalink):

1, 2, 12, 24, 36, 72, 240, 480, 720, 1440, 3360, 4320, 5280, 6240, 6720, 8160, 9120, 10080, 11040, 13440, 13920, 14880, 15840, 17760, 18720, 19680, 20160, 20640, 21600, 22560, 24480, 25440, 27360, 28320, 29280, 32160, 33120, 34080, 35040, 37920, 39840

27360 factorises to \(2^5 \times 3^2 \times 5 \times 19 \) and thus it has 72 divisors. These are:

1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 16, 18, 19, 20, 24, 30, 32, 36, 38, 40, 45, 48, 57, 60, 72, 76, 80, 90, 95, 96, 114, 120, 144, 152, 160, 171, 180, 190, 228, 240, 285, 288, 304, 342, 360, 380, 456, 480, 570, 608, 684, 720, 760, 855, 912, 1140, 1368, 1440, 1520, 1710, 1824, 2280, 2736, 3040, 3420, 4560, 5472, 6840, 9120, 13680, 27360

The number of divisors of these divisors is listed below and all these numbers divide 27360 as well:

1, 2, 2, 3, 2, 4, 4, 3, 4, 6, 4, 5, 6, 2, 6, 8, 8, 6, 9, 4, 8, 6, 10, 4, 12, 12, 6, 10, 12, 4, 12, 8, 16, 15, 8, 12, 6, 18, 8, 12, 20, 8, 18, 10, 12, 24, 12, 16, 24, 16, 12, 18, 30, 16, 12, 20, 24, 24, 36, 20, 24, 24, 32, 30, 24, 36, 40, 36, 48, 48, 60, 72

It will be 960 days before I see the next strongly refactorable number (28320). That will occur on October 16th, 2026.

ADDENDUM:

Today I reached the strongly refactorable number 28320. They occur about every three years so another three years till I reach 29280.

Tau Numbers

Having just posted about anti-tau numbers, I realised that I hadn't yet made a dedicated post about tau numbers and in fact only mentioned them briefly in a post titled Arithmetic Numbers. In this post, I'll address that deficiency. 

Wikipedia has the following definition:

A refactorable number or tau number is an integer \(n\) that is divisible by the count of its divisors, or to put it algebraically, \(n\) is such that \( \tau (n) \mid n \). The first few refactorable numbers are listed in OEIS A033950 as:

1, 2, 8, 9, 12, 18, 24, 36, 40, 56, 60, 72, 80, 84, 88, 96, 104, 108, 128, 132, 136, 152, 156, 180, 184, 204, 225, 228, 232, 240, 248, 252, 276, 288, 296, ...

For example, 18 has 6 divisors (1 and 18, 2 and 9, 3 and 6) and is divisible by 6. There are infinitely many refactorable numbers.

Up to 100,000, there are 5257 tau numbers representing 5.257% of the range. However, the article points out that these numbers have a natural density of zero. Another Wikipedia article explains what is meant by this term:

In number theory, natural density (also referred to as asymptotic density or arithmetic density) is one method to measure how "large" a subset of the set of natural numbers is. It relies chiefly on the probability of encountering members of the desired subset when combing through the interval [1, \(n\)] as \(n\) grows large.

Intuitively, it is thought that there are more positive integers than perfect squares, since every perfect square is already positive, and many other positive integers exist besides. However, the set of positive integers is not in fact larger than the set of perfect squares: both sets are infinite and countable and can therefore be put in one-to-one correspondence. Nevertheless if one goes through the natural numbers, the squares become increasingly scarce. The notion of natural density makes this intuition precise for many, but not all, subsets of the naturals (see Schnirelmann density, which is similar to natural density but defined for all subsets of \( \mathbb{N} \).

If an integer is randomly selected from the interval [1, \(n\)], then the probability that it belongs to A is the ratio of the number of elements of A in [1, \(n\)] to the total number of elements in [1,\( n\)]. If this probability tends to some limit as \(n\) tends to infinity, then this limit is referred to as the asymptotic density of A. This notion can be understood as a kind of probability of choosing a number from the set A. Indeed, the asymptotic density (as well as some other types of densities) is studied in probabilistic number theory.

It has been proven that there are no three consecutive integers that are all tau numbers. They can appear in pairs however, although not often. For example, up to 100,000, there are only 13 such pairs. These are:

(1, 2), (8, 9), (1520, 1521), (50624, 50625), (62000, 62001), (103040, 103041), (199808, 199809), (221840, 221841), (269360, 269361), (463760, 463761), (690560, 690561), (848240, 848241), (986048, 986049)

Take the last pair as an example:

\(986048 = 2^6 \times 7 \times 31 \times 71\) with \(56\) divisors such that \( 56 \mid 986048= 17608\)

\(986049 = 3^2 \times 331^2\) with \(9\) divisors such that \(9 \mid 986049= 109561\)

Whether there are an infinite number of such pairs is not known. Numbers Aplenty states that the smallest Pythagorean triple of tau numbers is (40, 96,104) which is not a primitive triple because it is a multiple of (5, 12, 13).  No one knows if there is a primitive triple.

Up to one million, there are 60 palindromic tau numbers. They are:

[1, 2, 8, 9, 88, 232, 252, 424, 444, 636, 808, 828, 2772, 4224, 12321, 21512, 21612, 23032, 23832, 24642, 25352, 25452, 27372, 29292, 40104, 40904, 42324, 42424, 42624, 44244, 46164, 46264, 46464, 48084, 48384, 48584, 48684, 61416, 61816, 63036, 63636, 65856, 67476, 67576, 69396, 69896, 80508, 82428, 84248, 84948, 86168, 86868, 88188, 88488, 216612, 270072, 423324, 426624, 468864, 486684]

Again, up to one million, there are also 2731 non-palindromic tau numbers whose reversals are also tau numbers. The first such number is 80 with reversal 8. The initial members of this sequence are:

80, 276, 288, 468, 480, 672, 864, 880, 882, 1440, 1656, 2000, 2025, 2148, 2160, 2176, 2178, 2196, 2320, 2388, 2700, 2988, 4044, 4050, 4068, 4080, 4240, 4284, 4404, 4668, 4824, 4856, 4860, 4896, 5202, 5220, 6561, 6584, 6712, 6720, 6912, 6984, 8080, 8100, 8412, 8604, 8649, 8664, 8712, 8832, 8892, 9468, 10000, ... permalink

It should be noted that \( \tau(n)=\sigma(n,0)\) and so this function can be used as an alternative to len(divisors(\(n\))) in any calculations.

There's an interesting history associated with the term refactorable number. To quote again from the Wikipedia article:

First defined by Curtis Cooper and Robert E. Kennedy where they showed that the tau numbers have natural density zero, they were later rediscovered by Simon Colton using a computer program he had made which invents and judges definitions from a variety of areas of mathematics such as number theory and graph theory. Colton called such numbers "refactorable". While computer programs had discovered proofs before, this discovery was one of the first times that a computer program had discovered a new or previously obscure idea. Colton proved many results about refactorable numbers, showing that there were infinitely many and proving a variety of congruence restrictions on their distribution. Colton was only later alerted that Kennedy and Cooper had previously investigated the topic.

Note that \(\tau\) is sometimes used to refer to \(2 \times \pi\) but that usage has nothing to do with this post. 

Anti-tau Numbers

I was familiar with what a tau number was, the definition being that it is an integer divisible by its number of its divisors.  Tau is the divisor function and returns the number of divisors of an integer e.g. \( \tau(6)=4\) because 6 has four divisors (1, 2, 3 and 6). However, 6 is not a tau number because 4 does not divide 6 without remainder.

In general, we can say that a number \(n\) is a tau number if \( \tau(n)|n\). 12 is a tau number because \( \tau(12) =6\) and \(6|12=2\). An alternative term, refactorable number, can be used for tau numbers (see Wikipedia entry). What then is an anti-tau number? Well, it turns out to be a number \(n\) such that \( \text{gcd}(n,\tau(n))=1\). These numbers comprise OEIS A046642.

A046642

Numbers \(k\) such that \(k\) and number of divisors \( \text{d}(k)\) are relatively prime.

Note that \( \text{d}(k)\) is sometimes used in place of \( \tau(k) \). All odd prime numbers will be anti-tau numbers because they have two divisors and 2 does not divide them. An example of an anti-tau number is 15 because it has four divisors (1, 3, 5 and 15) and gcd(15, 4) = 1. These numbers are quite frequent. There are 18080 in the range up to 40,000, representing over 45% of all the numbers in the range. The initial members are:

1, 3, 4, 5, 7, 11, 13, 15, 16, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 47, 49, 51, 53, 55, 57, 59, 61, 64, 65, 67, 69, 71, 73, 77, 79, 81, 83, 85, 87, 89, 91, 93, 95, 97, 100, 101, 103, 105, 107, 109, 111, 113, 115, 119, 121, 123, 125, 127, 129, 131

Notice that all even anti-tau numbers are square numbers (4, 16, 64, 100, 196, 256, 484, 676, 784, 1024 etc.) of the form \(4k^2\) where \(k\) is a positive even integer. However, not all even values of \(k\) produce anti-tau numbers. The initial values of \(k\) are 2, 4, 8, 10, 14, 16,  22, ...

By contrast, tau numbers are less common. In the range up to 100,000, there are 5257 of them representing 5.257% of the range. The anti-tau numbers were brought to my attention by one of the properties of the number representing my diurnal age, 26895, qualifying it for membership of OEIS A341780:

A341780

Starts of runs of 3 consecutive anti-tau numbers (A046642).

Let's look at 26895, 26896 and 26897 to confirm that this is true:

• \(26895 = 3 \times 5 \times 11 \times 163\) with 16 divisors and 16 is relatively prime to it
• \(26896 = 2^4 \times 41^2\) with 15 divisors and 15 is relatively prime to it
• \(26897 = 13 \times 2069\) with 4 divisors and 4 is relatively prime to it

Despite how common anti-tau numbers are, these runs of three (the maximum possible) are not that frequent. The problem is that the runs are of the form odd-even-odd and the even numbers must be of the form \(4k^2\) as mentioned earlier. It is these relatively sparse even anti-tau numbers that make these runs less common than might be expected. Notice how \(26896 = 4 \times 82^2\). There are only 69 such numbers in the range up to 100,000. Here is the list:

3, 15, 195, 255, 483, 783, 1023, 1155, 1295, 1443, 1599, 2703, 3363, 4623, 4899, 5183, 6399, 6723, 7395, 7743, 8463, 8835, 10815, 11235, 11663, 12099, 12543, 15375, 16383, 16899, 17955, 18495, 20163, 24963, 25599, 26895, 27555, 31683, 33855, 35343, 36099, 37635, 38415, 40803, 44943, 45795, 46655, 47523, 52899, 53823, 55695, 56643, 61503, 62499, 64515, 65535, 70755, 71823, 73983, 80655, 81795, 82943, 85263, 87615, 88803, 91203, 92415, 94863, 96099

Here is a permalink that will confirm all the above results.

Arithmetic Numbers

I was familiar with the concept of an arithmetic derivative but today I was reminded of the concept of an arithmetic number. I encounter the term almost daily when referring to the website Numbers Aplenty but hadn't paid it much attention. Here is a definition from Wikipedia:

In number theory, an arithmetic number is an integer for which the average of its positive divisors is also an integer. For instance, 6 is an arithmetic number because the average of its divisors is 3:$$ \frac{1+2+3+6}{4}=3$$Phrasing it more rigorously, we could say that a number \(n\) is arithmetic if the number of divisors \(d(n)\) or \( \sigma_0(n) \) divides the sum of divisors \( \sigma_1(n) \).

This ratio is not be confused with the ratio that I was investigating in my previous post on Barely Abundant Numbers. In that case, the ratio was \( \sigma(n) \) over \(n\) which determines whether a number is deficient, perfect or abundant. If the ratio is less than 2, the number is deficient. If it's greater than 2, the number is abundant. We know 6 is a perfect number because the ratio equals 2:$$ \frac{1+2+3+6}{6}=2$$The term was brought to my attention by one of the properties of the number associated with my diurnal age of 26744. The property qualifies the number for membership in OEIS A107924 as the 119th member.

A107924

Even numbers \(n\) such that \(n^2\) is an arithmetic number.

296, 536, 632, 872, 1208, 1304, 1544, 2072, 2216, 2648, 2984, 3584, 3656, 3752, 3848, 3896, 3904, 3992, 4328, 4424, 4568, 4904, 5624, 5672, 5912, 6008, 6104, 6584, 6968, 7016, 7256, 7352, 7928, 8216, 8264, 8456, 8696, 8896, 8936, 9032, 9128, 9176, 9368, 9608, 9704, 10184, 10376, 10616, 10808, 11048, 11336, 11384, 11624, 12008, 12392, 12632, 12728, 12968, 13304, 13504, 13976, 14072, 14312, 14408, 14472, 14648, 14984, 15512, 15704, 15992, 16088, 16424, 16568, 16616, 16664, 16952, 17064, 17096, 17432, 17768, 18008, 18056, 18344, 18536, 18776, 19016, 19112, 19592, 19784, 19832, 20024, 20072, 20456, 20888, 21368, 21464, 21608, 21704, 22136, 22376, 22808, 22952, 23048, 23384, 23488, 23872, 24152, 24392, 24488, 24776, 25496, 25592, 25736, 25832, 26072, 26168, 26408, 26504, 26744

The OEIS comments state that "odd numbers with this property are much more numerous" and this is indeed true. In the same range (up to 26744) there are 2462 such odd numbers. Checking for \(n=26744\), we find that \( n^2=715241536 \) and we have: $$ \frac{\sigma(715241536,1)}{\sigma(715241536, 0)}=\frac{1419732111}{21}=67606291$$It can be noted that 26744 is itself an arithmetic number but this is hardly surprising as most numbers are and this includes all odd primes. To see this, take any odd prime \(p\) with the sum of its two divisors being \(1+p\), an even number and thus divisible by 2. According to Numbers Aplenty, \(p^p\) is also an arithmetic number.

It is possible to find quite long runs of consecutive arithmetic numbers. For example, one of length 105 starts at 3033935561. The smallest 3 × 3 magic square made of consecutive arithmetic numbers is shown below:

The 10000th arithmetic number is 12953. This gives a good idea of their frequency and why numbers that aren't arithmetic numbers are more interesting because they are less frequent. For example, \(84=2^2 \times 3 \times 7\) has twelve divisors 1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, 84 that sum to 224. It is not an arithmetic number because 12 does not divide 224 without remainder.

Just to confuse matters, \(26744 = 2^3 \times 3343\) is a tau number, defined as a number that is divisible by the number of its divisors (in this case, eight). If we put all this together and ask what even numbers \(n\) have the properties that:$$ \begin{align} \frac{ \sigma(n^2,1)}{\sigma(n^2,0)}&=k_1\\ \frac{ \sigma(n,1)}{\sigma(n,0)}&=k_2\\ \frac{ n}{\sigma(n,0)}&=k_3 \end{align}$$where \(k_1, k_2, k_3\) are integers. It turns that the list is fairly short. In the range up to 40000, there are only 29 such numbers: 

1, 632, 1208, 3896, 6008, 6584, 7352, 7928, 8696, 10616, 11384, 13304, 14072, 14648, 15992, 20024, 21368, 22136, 26168, 26744, 28088, 28856, 30776, 31544, 32888, 35576, 36152, 38264, 39608

Thue-Morse Constant

On June 20th 2020, I made a post titled Prouhet-Thue-Morse Sequence named for Eugène Prouhet, Axel Thue, and Marston Morse (the Prouhet reference is sometimes omitted). By the way, the Thue part is named after Axel Thue, whose name is pronounced as if it were spelled "Tü" where the ü sound is roughly as in the German word üben. It is incorrect to say "Too-ee" or "Too-eh". Thus sayeth N. J. A. Sloane, June 12th 2018, in his comments about OEIS A010060 that lists the members of the sequence. 

It is a most interesting sequence and my blog post covers it quite well and has links to three interesting YouTube videos. However, there is a so-called Thue-Morse constant that is the topic of this post. The sequence begins:

 0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 1, 0, 1, 1, 0, 0, ... 

If we concatenate these binary digits, we get a binary number:

 \(P=0.0110100110010110100101100..._2\) 

This number can be converted a decimal and is represented by the Greek letter \( \tau \):$$\tau=\sum_{n=0}^{\infty} \frac{t_i}{2^{i+1}}=0.4124540336401075977 \dots$$where \(t_i\) is the \(i^{th}\) element of the binary Thue-Morse sequence. The number has been shown to be transcendental.

Figure 1 provides two interesting expressions for the Thue-Morse constant (source):

Figure 1

I came across the constant by means of my diurnal age investigation, discovering that the number associated with my diurnal age (26547) was a member of OEIS A096394:

A096394

Engel expansion of Thue-Morse constant.

The sequence begins 3, 5, 6, 9, 12, 19, 92, 173, 242, 703, 1861, 3186, 4746, 7843, 26547, ... and the comments state that:$$ 0.4124540336 \dots = \frac{1}{3}+\frac{1}{3 \times 5}+\frac{1}{3 \times 5 \times 6}+\frac{1}{3 \times 5 \times 6 \times 9} + \dots$$I made a post about Engel Expansions way back on September 28th 2016.

If we take 0.412454033640107597783361368258455283089 as an approximation of \(\tau\) and plug this into the SageMathCell formula listed in this post, we do confirm that 26547 is a member. To generate further members of the sequence however, the number of decimal places to which \( \tau \) needs to be approximated must be increased. Here is a permalink to SageMathCell while the code is listed below (blue for input and red for output).

x=0.412454033640107597783361368258455283089

u=x

E=[1]

F=[]

product=1

sum=0

for i in [1..15]:

    a=ceil(1/u)

    u=u*a-1

    E.append(a)

    product=product*a

    sum+=1/product

    F.append(1/product)

print(E, "... this is the Engels expansion")

[1, 3, 5, 6, 9, 12, 19, 92, 173, 242, 703, 1861, 3186, 4746, 7843, 26547] ... this is the Engels expansion