Pronic Determinants of Circulant Matrices

Consider the number 28235 that is my diurnal age today. It has a circulant matrix with a determinant 10100 that is a pronic number since 10100 = 100 x 101.$$\begin{bmatrix}

2 & 8 & 2 & 3 & 5 \\

8 & 2 & 3 & 5 & 2 \\

2 & 3 & 5 & 2 & 8 \\

3 & 5 & 2 & 8 & 2 \\

5 & 2 & 8 & 2 & 3

\end{bmatrix}$$What's interesting is that most of the permutations of the digits of 28235 have determinants of their circulant matrices that are also pronic (permalink).

Number       | Determinant     | Factorisation  
------------------------------------------------
22358        | 15500           | 124 x 125      
22385        | 10100           | 100 x 101      
22538        | 19100           |                
22583        | 10100           | 100 x 101      
22835        | 19100           |                
22853        | 15500           | 124 x 125      
23258        | 19100           |                
23285        | 19100           |                
23528        | 10100           | 100 x 101      
23582        | 15500           | 124 x 125      
23825        | 15500           | 124 x 125      
23852        | 10100           | 100 x 101      
25238        | 15500           | 124 x 125      
25283        | 15500           | 124 x 125      
25328        | 10100           | 100 x 101      
25382        | 19100           |                
25823        | 19100           |                
25832        | 10100           | 100 x 101      
28235        | 10100           | 100 x 101      
28253        | 10100           | 100 x 101      
28325        | 15500           | 124 x 125      
28352        | 19100           |                
28523        | 19100           |                
28532        | 15500           | 124 x 125      
32258        | 10100           | 100 x 101      
32285        | 15500           | 124 x 125      
32528        | 15500           | 124 x 125      
32582        | 19100           |                
32825        | 10100           | 100 x 101      
32852        | 19100           |                
35228        | 19100           |                
35282        | 10100           | 100 x 101      
35822        | 15500           | 124 x 125      
38225        | 19100           |                
38252        | 15500           | 124 x 125      
38522        | 10100           | 100 x 101      
52238        | 10100           | 100 x 101      
52283        | 19100           |                
52328        | 19100           |                
52382        | 15500           | 124 x 125      
52823        | 10100           | 100 x 101      
52832        | 15500           | 124 x 125      
53228        | 15500           | 124 x 125      
53282        | 10100           | 100 x 101      
53822        | 19100           |                
58223        | 15500           | 124 x 125      
58232        | 19100           |                
58322        | 10100           | 100 x 101      
82235        | 15500           | 124 x 125      
82253        | 19100           |                
82325        | 19100           |                
82352        | 10100           | 100 x 101      
82523        | 15500           | 124 x 125      
82532        | 10100           | 100 x 101      
83225        | 10100           | 100 x 101      
83252        | 15500           | 124 x 125      
83522        | 19100           |                
85223        | 10100           | 100 x 101      
85232        | 19100           |                
85322        | 15500           | 124 x 125  

Note that it is only when the determinant is 19100 that it is not pronic since 19100 = 100 x 191. In my post titled Determinants of Circulant Matrices, I listed all numbers up to 40000 with the property that the determinants of their circulant matrices were pronic. The numbers between 28000 and 40000 are:

28235, 28253, 28325, 28327, 28453, 28479, 28532, 28543, 28574, 28619, 28732, 28776, 29054, 29168, 29245, 29254, 29555, 29700, 29748, 30171, 30179, 30566, 30575, 30665, 30900, 31100, 31107, 31134, 31233, 31323, 31332, 31355, 31358, 31385, 31400, 31413, 31422, 31440, 31510, 31637, 31646, 31684, 31763, 31907, 32124, 32133, 32223, 32232, 32241, 32258, 32285, 32287, 32313, 32322, 32331, 32528, 32728, 32825, 32845, 32854, 32960, 33123, 33132, 33141, 33176, 33213, 33222, 33231, 33312, 33321, 33335, 33353, 33515, 33518, 33533, 33569, 33671, 33789, 33815, 33965, 33987, 34100, 34166, 34212, 34258, 34311, 34410, 34582, 34599, 34700, 34861, 35153, 35183, 35248, 35282, 35333, 35482, 35507, 35531, 35606, 35693, 35822, 35831, 35936, 35949, 35996, 35999, 36056, 36065, 36137, 36359, 36395, 36418, 36461, 36506, 36614, 36713, 36920, 36995, 37011, 37055, 37091, 37316, 37361, 37400, 37700, 37799, 37822, 37893, 37938, 37979, 38146, 38153, 38252, 38272, 38379, 38397, 38425, 38522, 38524, 38531, 39495, 39536, 39569, 39599, 39653, 39659, 39738, 39797, 39873, 39900, 39954, 39959, 39977, 39995

Note that with 28235 and its digit permutations there are TWO determinants that satisfy. These are:

• \(10100 = 100 \times 101\)
• \(15500 = 124 \times 125\)

This is generally not the case. Consider 28327 and its digit permutations where only the determinant 14762 = 121 x 122 satisfies (permalink).

Number       | Determinant     | Factorisation  
------------------------------------------------
22378        | 14762           | 121 x 122      
22387        | 11462           |                
22738        | 27962           |                
22783        | 11462           |                
22837        | 27962           |                
22873        | 14762           | 121 x 122      
23278        | 27962           |                
23287        | 27962           |                
23728        | 11462           |                
23782        | 14762           | 121 x 122      
23827        | 14762           | 121 x 122      
23872        | 11462           |                
27238        | 14762           | 121 x 122      
27283        | 14762           | 121 x 122      
27328        | 11462           |                
27382        | 27962           |                
27823        | 27962           |                
27832        | 11462           |                
28237        | 11462           |                
28273        | 11462           |                
28327        | 14762           | 121 x 122      
28372        | 27962           |                
28723        | 27962           |                
28732        | 14762           | 121 x 122      
32278        | 11462           |                
32287        | 14762           | 121 x 122      
32728        | 14762           | 121 x 122      
32782        | 27962           |                
32827        | 11462           |                
32872        | 27962           |                
37228        | 27962           |                
37282        | 11462           |                
37822        | 14762           | 121 x 122      
38227        | 27962           |                
38272        | 14762           | 121 x 122      
38722        | 11462           |                
72238        | 11462           |                
72283        | 27962           |                
72328        | 27962           |                
72382        | 14762           | 121 x 122      
72823        | 11462           |                
72832        | 14762           | 121 x 122      
73228        | 14762           | 121 x 122      
73282        | 11462           |                
73822        | 27962           |                
78223        | 14762           | 121 x 122      
78232        | 27962           |                
78322        | 11462           |                
82237        | 14762           | 121 x 122      
82273        | 27962           |                
82327        | 27962           |                
82372        | 11462           |                
82723        | 14762           | 121 x 122      
82732        | 11462           |                
83227        | 11462           |                
83272        | 14762           | 121 x 122      
83722        | 27962           |                
87223        | 11462           |                
87232        | 27962           |                
87322        | 14762           | 121 x 122   

Artin's Constant

My diurnal age today is 28229, a prime, with a reciprocal that has a period of 28228. It is thus one of those primes \(p\) with a period that is given by \( \dfrac{1}{p-1}\).

I got to wondering what proportion of primes have reciprocals that are one less than the prime itself. I asked Gemini. It's response follows and comprises the rest of this post.

Approximately 37.3956% of all prime numbers have reciprocals with a repeating decimal period of maximum length, \(p - 1\). In number theory, primes with this property are known as full reptend primes (or long primes) in base 10.

The Number Theory Behind Maximal Periods

The number of digits in the repeating period of the reciprocal \(1/p\) is determined by the multiplicative order of \(10\) modulo \(p\), denoted as \(\text{ord}_p(10)\).

By Fermat's Little Theorem, for any prime \(p \neq 2, 5\), the period length \(k\) is the smallest positive integer such that:

\[10^k \equiv 1 \pmod{p}\]

Since the multiplicative group \((\mathbb{Z}/p\mathbb{Z})^\times\) has order \(\phi(p) = p - 1\), Lagrange's theorem dictates that the period length \(k\) must always divide \(p - 1\).

For the period to achieve the absolute maximum length of \(p - 1\), the integer \(10\) must be a primitive root modulo \(p\). When \(10\) is a primitive root, the successive powers of \(10 \pmod{p}\) cycle through every possible non-zero remainder from \(1\) to \(p - 1\) before repeating, generating a cyclic period of length \(p - 1\).

---

Artin's Constant and Asymptotic Density

The exact asymptotic density of primes for which a given integer is a primitive root is governed by Artin's conjecture on primitive roots.

Because the base \(a = 10\) is neither a perfect square nor \(-1\), Artin's conjecture predicts that the proportion of full reptend primes among all primes up to \(N\) converges to a universal mathematical constant as \(N \to \infty\). This value is known as Artin's constant (\(C_{\text{Artin}}\)), defined by the infinite product over all prime numbers \(q\):

\[C_{\text{Artin}} = \prod_{q \text{ prime}} \left(1 - \frac{1}{q(q-1)}\right) \approx 0.3739558136\dots\]

While Emil Artin's conjecture remains unconditionally unproven for specific bases like 10, Christopher Hooley proved in 1967 that the asymptotic density is exactly \(C_{\text{Artin}}\) under the assumption of the Generalized Riemann Hypothesis (GRH). Extensive computational verification confirms that the distribution stabilizes at this ~37.3956% threshold (cataloged as decimal expansion A005596 in the OEIS).

---

Period Lengths of Small Primes

Comparing the first few primes highlights how the multiplicative order behaves when \(10\) is (and is not) a primitive root:

Prime (\(p\))	Reciprocal (\(1/p\))	Period Length (\(\text{ord}_p(10)\))	Full Reptend? (\(k = p - 1\))
3	\(0.\overline{3}\)	\(1\)	No
7	\(0.\overline{142857}\)	\(6\)	Yes
11	\(0.\overline{09}\)	\(2\)	No
13	\(0.\overline{076923}\)	\(6\)	No
17	\(0.\overline{0588235294117647}\)	\(16\)	Yes
19	\(0.\overline{052631578947368421}\)	\(18\)	Yes
23	\(0.\overline{0434782608695652173913}\)	\(22\)	Yes

The sequence of base-10 full reptend primes continues: \(7, 17, 19, 23, 29, 47, 59, 61, 97, 109, 113, 131, 149, \dots\) (OEIS sequence A001122). Noticeably, primes like \(13\) (where \(10^6 \equiv 1 \pmod{13}\)) and \(37\) (where \(10^3 \equiv 1 \pmod{37}\)) fall into sub-cyclic periods because their multiplicative orders are proper divisors of \(p - 1\).

Loeschian Numbers Revisited

 I've blogged about Loeschian numbers in the following posts:

• Loeschian Numbers on the 5th January 2022
• Loeschian Primes on the 21st May 2023
• Linear Relationships Between Loeschian Numbers on the 11th June 2023

Here is some additional information provided by Gemini:

Approximately 19.63% of the positive integers between 1 and 40000 are Loeschian numbers. This equates to roughly 7851 integers in that range.

Why the Percentage Drops Over Time? You might intuitively expect numbers representable by $i^2 + i \, j + j^2$ to maintain a steady density, but they actually become progressively sparser as you move higher up the number line.

In 1975, economic geographer J. U. Marshall tabulated that between 1 and 10000 inclusive, there are exactly 2299 Loeschian numbers—a density of 22.99% [7.1.5]. By the time you reach 40000, that density drops by over three percentage points.

This thinning behavior is governed by a Landau-Ramanujan-like asymptotic law. The total count of Loeschian numbers $N(x)$ up to a threshold $x$ grows according to the formula:$$N(x) \sim \alpha \frac{x}{\sqrt{\ln x}}$$

Here, $\alpha$ is the Loeschian density constant (OEIS A301429), which is approximately $0.6389094$. Because the denominator $\sqrt{\ln x}$ grows continuously as $x$ increases, the overall percentage of Loeschian numbers steadily decays toward zero as you approach infinity.

The total Loeschian numbers up to 1000000 is 180874, a percentage 18.09% (permalink).

I've now incorporated whether a number is Loeschian or not into my daily number analysis (permalink) as shown below where \(28225=15^2+15 \times 160 + 160^2\).

28224: An Interesting Number

28224 has 194 entries in the OEIS which is extraordinarily high for a five digit number. In this post I'll be discussing some of this number's most interesting properties but not all of them. There are just too many. It's prime factorisation is:$$28224=2^6 \times 3^2 \times 7^2$$FIRST INTERESTING PROPERTY

Numbers that are perfect squares are quite rare in the range up to 40000. There are only 200 of them and 28224, my diurnal age today, is one of them. It has the property that:$$28224=168^2$$The number of days between my experience of them is a little less than a year. There is a gap of exactly 365 days between \(183^2\) and \(182^2\) since:$$ \begin{align} 183^2-182^2 &= (183 + 182)(183-182) \\ &=365 \times 1 \\ &=365 \end{align}$$I'll be \(33124\) or \(182^2\) days when I'm over \(90\) years old so I may not get to experience the transition from this square to the next.

28224 is also a Loeschian number since it is equal to \(72^2+ 72 \times 120 + 120^2\).

28224 also has a product of digits (256) that is a perfect square since \(256=16^2\).

SECOND INTERESTING PROPERTY

Numbers that are the sum of two positive cubes are relatively rare in the range up to 40000. In fact, there are only 378 numbers in the range up to 40000 and 28224 is one of them because:$$28224=22^3 + 26^3$$These numbers form OEIS A004999.

THIRD INTERESTING NUMBER

Energetic numbers are numbers that can be broken into two or more substrings and expressed as a sum of (possibly different) positive powers of those substrings. They form OEIS  A055480. 28224 is one such number because:$$28224=28^3 + 2^{11} + 2^7 + 4^6$$I discuss this category of numbers in my blog post Energetic Numbers.

FOURTH INTERESTING NUMBER

Friedman numbers are positive integers which can be written in some non-trivial way using its own digits, together with the symbols + – × / ^ ( ) and concatenation. 28224 is one such number because:$$28224 = (2 + 82)^2 × 4$$It is said to be a "nice" Friedman number because the digits are in the same order as the number. These numbers are listed on my blog post Narcissistic, D-Powerfull and Friedman Numbers.

FIFTH INTERESTING NUMBER

28224 has the property that certain of its factors (not necessarily prime) can be arranged to form a palindrome. Specifically:$$2 \times 2 \times 2 \times 882 \times 2 \times 2 = 22288222$$I've written about these sorts of numbers in a post titled Why Is 313131 An Interesting Number?

SIXTH INTERESTING PROPERTY

28224 is a concatenation of powers of 2 since:$$28224= 2^1 \; || \; 2^3 \; || \; 2^1 \; || \; 2^1 \; || \; 2^2$$I've written about numbers that can be formed in this way in a blog post titled Nothing New Under The Sun. It is also a concatenation of multiples of 7 since:$$28224= (7 \times 4) \, || \, (7 \times 32)$$I posted about these sorts of concatenations in my blog post More Numbers as Concatenations.

SEVENTH INTERESTING PROPERTY

28224 is a member of OEIS A253824 where$$ \text{numbers } m = s \, || \, t \text{ such that } m = \sigma(s) \times \sigma(t)$$where || represents concatenation. In the case of 28224 we have:$$ \begin{align} 28224 &= 28 \, || \, 224 \\ &= \sigma(28) \times \sigma(224) \\ &= 56 \times 504 \\ &=28224 \end{align}$$28224 is only the third such number in the range up to 40000. The two earlier numbers are 540 and 2352.

EIGHTH INTERESTING PROPERTY

28224 has a digit sum of 18 and when this is added to the number the result is 28242 which has the same digits as 28224 but in a slightly different order. This property makes it a member of OEIS A246420.

A246420

Numbers \(n\) such that \(n\)  + digit sum of \(n\) is a permutation of the decimal digits of \(n\) .

Revisiting the Divisors Algorithm

 Figure 1 shows the Divisors Algorithm that I developed early this year (link):

Figure 1

Applied to the number associated with my diurnal age today, 28223, we get the following trajectory:

28223, 169338, 4064112, 365770080, 1016028, 54865512, 7900633728, 20574567, 864131814, 165913308288, 192029292, 41478327072, 64009764, 11521757520, 13716378, 987579216, 237019011840, 243846720, 81932497920, 108150897254400, 17882092800, 17381394201600, 5486551200, 8466900, 914425200, 2032056, 28223

Figure 2 shows the trajectory. Note that the sequence returns to its starting point.

Figure 2

The vertical scale is logarithmic and the maximum value reached is impressive. Not every number enters a loop that returns it to its starting point. 28222 has a trajectory that enters a loop but it does not return to its starting point. The loop is reached at a value that is twice that of the starting point: 56444 = 2 x 28222. The trajectory is:

28222, 225776, 4515520, 252869120, 1580432, 63217280, 493885, 7902160, 98777, 790216, 25286912, 1820657664, 6321728, 112888, 1806208, 56444, 677328, 27093120, 211665, 3386640, 42333, 338664, 10837248, 780281856, 4063968, 56444

Figure 3 shows the trajectory:

Figure 3

Other numbers may or may not enter a loop but we have to call a stop somewhere. 28237 is an example of such a number where we call a halt after 99 steps. The trajectory is as shown in Figure 4 and it seems to be heading for the stars but who knows?

Figure 4

Once again, this algorithm is base-independent and so the trajectories will be identical regardless of the number base used. The trend of the trajectory for all numbers is generally upwards because of the primes \(p \rightarrow 2p+1\) but the situation will be different if we modify the rule for the primes as shown below. 

\(\text{For } p \text{ prime:}\)$$ \begin{align} &p \rightarrow 2p+1 \text{ if } p \! \bmod 4 \equiv 1 \\ &p \rightarrow \frac{p-1}{2} \text{ if }  p \bmod 4 \equiv 3 \end{align}$$

It would be interesting to investigate the trajectories using this modification. If no primes are encountered then the trajectories of course will be identical (as is the case with 28237 mentioned earlier). Here is a permalink to an implementation of this modification for the cases of:

• primes \(p\) as above and composites \(n\) with factors counted \( \textbf{with} \) multiplicity. If \(f\) is the number of factors then:
• if \( n \bmod f \equiv 0 \) then \(n \rightarrow n/f\)
• if \( n \bmod f \not\equiv 0 \) then \(n \rightarrow n\times f\)
  
• primes as above and composites with factors counted \textbf{without}\) multiplicity. If \(f\) is the number of factors then:
• if \( n \bmod f \equiv 0 \) then \(n \rightarrow n/f\)
• if \( n \bmod f \not\equiv 0 \) then \(n \rightarrow n\times f\)
  
• primes as above and composites with divisors counted. If \(d\) is the number of divisors then:
• if \( n \bmod d \equiv 0 \) then \(n \rightarrow n/d\)
• if \( n \bmod d \not\equiv 0 \) then \(n \rightarrow n\times d\)

Here is the output for the number 28224:

=========================================

 Trajectory Analysis for N = 28224 

=========================================

1. Rule: Number of Factors (With Multiplicity)

   Highest Value: 2328480

   Trajectory Length: 15 steps

   Full Path: [28224, 282240, 23520, 211680, 2328480, 194040, 21560, 3080, 18480, 2310, 462, 1848, 308, 77, 154, 462]

   >>> Loop Detected: The sequence loops back to 462

--------------------------------------------------

2. Rule: Number of Factors (Without Multiplicity)

   Highest Value: 28224

   Trajectory Length: 9 steps

   Full Path: [28224, 9408, 3136, 1568, 784, 392, 196, 98, 49, 49]

   >>> Loop Detected: The sequence loops back to 49

--------------------------------------------------

3. Rule: Total Number of Divisors

   Highest Value: 28224

   Trajectory Length: 12 steps

   Full Path: [28224, 448, 32, 192, 2688, 84, 7, 3, 1, 2, 5, 11, 5]

   >>> Loop Detected: The sequence loops back to 5

--------------------------------------------------

More On The RDIV Algorithm

Under the RDIV or Recurring Digital Invariant Variant (a weird name I know) Algorithm, all numbers that are not narcissistic will enter a loop or terminate in a narcissistic number. I was interested in the proportion of numbers that terminate in a narcissistic number and so I had Gemini create an algorithm (permalink) to determine this. In the range up to 40000, there are 12224 such numbers which account for 30.56% of the range. These include the few numbers that are narcissistic themselves namely 0, 1, 2, 3, 4, 5, 6, 7, 8, 9,153, 370, 371, 407, 1634, 8208 and 9474 within the range. 

Here is a fuller list of narcissistic numbers (OEIS A005188):

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 153, 370, 371, 407, 1634, 8208, 9474, 54748, 92727, 93084, 548834, 1741725, 4210818, 9800817, 9926315, 24678050, 24678051, 88593477, 146511208, 472335975, 534494836, 912985153, 4679307774, 32164049650, 32164049651, 40028394225, 42678290603

While 12224 numbers are far too numerous to list here, we can thin the numbers by considering only triplets - meaning groups of three consecutive numbers that all lead to narcissistic numbers. Take for example, the numbers 28220, 28221 and 28222. Let's look at  their trajectories under the RDIV algorithm (permalink for generation). It will be seen that all three terminate in narcissistic numbers.

==================================================
RDIV TRAJECTORY ANALYSIS FOR INPUT: 28220
==================================================
Full Trajectory Visited:
28220, 32864, 41843, 35060, 11144, 2051, 642, 288, 
1032, 98, 145, 190, 730, 370

Loop Entry Point:   370 (encountered at step 14)
Pre-period Length:  13 step(s) before entering cycle
Cycle Length:       1 distinct number(s) in the loop
Canonical Cycle:    370

==================================================
RDIV TRAJECTORY ANALYSIS FOR INPUT: 28221
==================================================
Full Trajectory Visited:
28221, 32865, 43944, 62364, 16851, 43671, 25851, 
39051, 62418, 41601, 8802, 8208

Loop Entry Point:   8208 (encountered at step 12)
Pre-period Length:  11 step(s) before entering cycle
Cycle Length:       1 distinct number(s) in the loop
Canonical Cycle:    8208
==================================================

==================================================
RDIV TRAJECTORY ANALYSIS FOR INPUT: 28222
==================================================
Full Trajectory Visited:
28222, 32896, 99868, 191410, 535540, 51700, 19933, 118585,
555540, 66596, 85502, 39050, 62417, 25640, 11957, 78983,
141635, 67108, 57352, 23332, 793, 1099, 13123, 520, 133, 
55, 50, 25, 29, 85, 89, 145, 190, 730, 370

Loop Entry Point:   370 (encountered at step 35)
Pre-period Length:  34 step(s) before entering cycle
Cycle Length:       1 distinct number(s) in the loop
Canonical Cycle:    370
==================================================

In the range up to 40000 there are 1786 such triplets and if we restrict the range to those above 28000, there are only 220 triplets. The central members of each triplet are listed below:

28134, 28221, 28314, 28365, 28563, 28635, 28653, 29121, 29211, 29278, 29728, 29729, 29792, 29972, 30006, 30051, 30055, 30060, 30061, 30151, 30160, 30221, 30222, 30223, 30224, 30233, 30234, 30242, 30249, 30250, 30251, 30252, 30280, 30323, 30324, 30343, 30422, 30433, 30501, 30505, 30510, 30511, 30520, 30521, 30522, 30561, 30601, 30610, 30651, 30820, 31051, 31060, 31111, 31112, 31113, 31114, 31115, 31132, 31133, 31142, 31143, 31284, 31312, 31313, 31412, 31413, 31474, 31475, 31501, 31510, 31744, 31745, 31824, 31839, 32021, 32022, 32023, 32024, 32033, 32034, 32042, 32049, 32050, 32051, 32052, 32080, 32184, 32200, 32201, 32202, 32203, 32204, 32221, 32246, 32254, 32303, 32304, 32402, 32409, 32410, 32426, 32453, 32501, 32502, 32519, 32524, 32529, 32543, 32649, 32685, 32800, 32814, 32865, 33023, 33024, 33043, 33112, 33113, 33199, 33203, 33204, 33310, 33332, 33403, 33556, 33564, 33573, 33574, 33654, 33753, 33754, 34022, 34033, 34112, 34113, 34174, 34175, 34202, 34209, 34210, 34226, 34253, 34303, 34470, 34482, 34523, 34629, 34714, 34715, 34809, 34842, 35001, 35005, 35010, 35011, 35020, 35021, 35022, 35061, 35101, 35110, 35201, 35202, 35219, 35224, 35229, 35243, 35356, 35364, 35373, 35374, 35423, 35500, 35536, 35557, 35558, 35564, 35565, 35601, 35634, 35654, 35655, 35733, 35734, 36001, 36010, 36051, 36249, 36285, 36354, 36429, 36501, 36534, 36554, 36555, 36740, 36825, 37144, 37145, 37353, 37354, 37414, 37415, 37533, 37534, 37640, 37898, 37988, 38020, 38124, 38139, 38214, 38265, 38409, 38442, 38625, 38798, 38978, 39788, 39878

What if we look for quadruplets of such numbers, that is four consecutive numbers such that each of them leads to a narcissistic number under the RDIV algorithm. There are 843 such quadruplets. Let's consider 29727, 29728, 29729 and 29730 (permalink for generation):

==================================================
RDIV TRAJECTORY ANALYSIS FOR INPUT: 29727
==================================================
Full Trajectory Visited:
29727, 92727

Loop Entry Point:   92727 (encountered at step 2)
Pre-period Length:  1 step(s) before entering cycle
Cycle Length:       1 distinct number(s) in the loop
Canonical Cycle:    92727
==================================================

==================================================
RDIV TRAJECTORY ANALYSIS FOR INPUT: 29728
==================================================
Full Trajectory Visited:
29728, 108688, 833089, 1057187, 3822365, 2459843, 
6993329, 14633345, 2220997, 10389865, 78677956, 
80868197, 100822787, 349143695, 787435454, 219637307, 
478245278, 351621107, 52404626, 3881441, 4229261, 
5079674, 6804500, 2471597, 6524693, 5439665, 5517662, 
1539794, 10486178, 41064387, 24359267, 50954372, 
49665125, 47252996, 93994532, 129609702, 825273306, 
186642546, 166928742, 582410385, 272624046, 61034823, 
18535747, 29160166, 48085827, 56552866, 22988195, 
120039012, 387441198, 696753525, 453809058, 660044022, 
20680704, 24287425, 23064482, 18594977, 114856422, 
146773863, 235381845, 272643729, 478488762, 493962894, 
1307101374, 566117175, 104768859, 708502518, 312695826, 
543767442, 93544662, 46934467, 52373923, 49222342, 
43185378, 39788517, 88527942, 82822627, 40999873, 
151754277, 125229729, 817149759, 991981191, 1683899688, 
11389527500, 42046156391, 32164049651

Loop Entry Point:   32164049651 (encountered at step 85)
Pre-period Length:  84 step(s) before entering cycle
Cycle Length:       1 distinct number(s) in the loop
Canonical Cycle:    32164049651
==================================================

==================================================
RDIV TRAJECTORY ANALYSIS FOR INPUT: 29729
==================================================
Full Trajectory Visited:
29729, 134969, 1114364, 314894, 802507, 395482, 814099, 
1329123, 4787601, 4040559, 4971987, 13326561, 3763237, 
1933711, 5610889, 9335335, 4947967, 11525728, 23323781, 
22562213, 2077827, 4568037, 3297327, 6434685, 2770104, 
1663599, 10206123, 1686691, 7719931, 11215213, 397702, 
767532, 298372, 912091, 1062948, 7176570, 2828691, 
9257466, 6261021, 560130, 63011, 8021, 4113, 339, 
783, 882, 1032, 98, 145, 190, 730, 370

Loop Entry Point:   370 (encountered at step 52)
Pre-period Length:  51 step(s) before entering cycle
Cycle Length:       1 distinct number(s) in the loop
Canonical Cycle:    370
==================================================

==================================================
RDIV TRAJECTORY ANALYSIS FOR INPUT: 29730
==================================================
Full Trajectory Visited:
29730, 76131, 24828, 66624, 24384, 35091, 62418, 41601, 
8802, 8208

Loop Entry Point:   8208 (encountered at step 10)
Pre-period Length:  9 step(s) before entering cycle
Cycle Length:       1 distinct number(s) in the loop
Canonical Cycle:    8208
==================================================

Because there are four consecutive numbers, only the first and smallest will be listed and so in this case the number would be 29727. In the range between 28000 and 40000, here are the 60 initial or smallest members of each quadruplet:

29727, 30059, 30220, 30221, 30222, 30232, 30248, 30249, 30250, 30322, 30509, 30519, 30520, 31110, 31111, 31112, 31113, 31131, 31141, 31311, 31411, 31473, 31743, 32020, 32021, 32022, 32032, 32048, 32049, 32050, 32199, 32200, 32201, 32202, 32302, 32408, 32500, 33022, 33111, 33202, 33572, 33752, 34111, 34173, 34208, 34713, 35009, 35019, 35020, 35200, 35372, 35556, 35563, 35653, 35732, 36553, 37143, 37352, 37413, 37532

As for quintuplets, there are 451 of them in the range up to 40000. However, above 28000 there are only 16 and the initial or smallest members of each are:

30220, 30221, 30248, 30249, 30519, 31110, 31111, 31112, 32020, 32021, 32048, 32049, 32199, 32200, 32201, 35019

There are eight sextuplets: 30220, 30248, 31110, 31111, 32020, 32048, 32199, 32200.

There are two septuplets: 31110 and 32199 and no octuplets within the range.

Recurring Digital Invariant Variant (RDIV) Algorithm

Let's consider the following algorithm (formally called the Recurring Digital Invariant Variant or RDIV algorithm - see this link for an explanation of the name):

• choose a number \(n\)
• let \(k\) be the number of digits in \(n\)
• raise each digit of \(n\) to the \(k\)-th power and add the results
• call the new number \(n\) and repeat

Let's use \(n=14\) as an example:

• \(14 \rightarrow 1^2 + 4^2 = 17\)
• \(17 \rightarrow 1^2 + 7^2 = 50\)
• \(50 \rightarrow 5^2 + 0^2 = 25\)
• \(25 \rightarrow 2^2 + 5^2 = 29\)
• \(29 \rightarrow 2^2 + 9^2 = 85\)
• \(85 \rightarrow 8^2 + 5^2 = 89\)
• \(89 \rightarrow 8^2 + 9^2 = 145\)
• \(145 \rightarrow 1^3 + 4^3 + 5^3 = 190\)
• \(190 \rightarrow 1^3 + 9^3 + 0^3 = 730\)
• \(730 \rightarrow 7^3 + 3^3 + 0^3 = 370\)
• \(370 \rightarrow 3^3 + 7^3 + 0^3 = 370\) 

370 is a narcissistic number as explained in my post Narcissistic, D-Powerfull and Friedman Numbers. The trajectory of any number under this algorithm will either end with a narcissistic number (as was the case with 14) or it will enter a loop (as is the case with 28218). The latter has the following trajectory (permalink):

==================================================

RDIV TRAJECTORY ANALYSIS FOR INPUT: 28218

==================================================

Full Trajectory Visited:

28218, 65601, 18678, 90120, 59082, 94974, 136953, 595181, 824837, 646826, 406272, 168529, 855931, 825565, 355739, 681798, 1220035, 80569, 102718, 379859, 1459029, 9660576, 6524445, 485466, 379273, 768261, 473170, 240124, 8321, 4194, 7074, 5058, 5346, 2258, 4753, 3363, 1539, 7268, 7809, 13058, 36137, 25070, 19964, 126899, 1371747, 2489202, 6896889, 16417266, 10869443, 61641187, 25966788, 86116067, 27580867, 47154531, 6683686, 5316235, 440689, 848433, 533938, 811397, 911965, 1125165, 436317, 169860, 886898, 1626673, 1665667, 2021413, 18829, 124618, 312962, 578955, 958109, 1340652, 376761, 329340, 537059, 681069 -> [loops back to 886898]

Loop Entry Point:   886898 (encountered at step 65)

Pre-period Length:  64 step(s) before entering cycle

Cycle Length:       14 distinct number(s) in the loop

Canonical Cycle:    18829, 124618, 312962, 578955, 958109, 1340652, 376761, 329340, 537059, 681069, 886898, 1626673, 1665667, 2021413

==================================================

Figure 1 shows a graph of its trajectory:

Figure 1: permalink

I've incorporated this algorithm into my daily number analysis.