A Formidable Run Of Hidden Beast Numbers

Today marks the start of a remarkable run of diurnal age numbers constituting what I call \( \textbf{hidden beast numbers} \) that I discussed in a post titled More Hidden Beast Numbers. These are numbers that contain the digit sequence "666" when expressed in a non-decimal base. In the case of my diurnal age today (28032) and the run of numbers to come, that base is 8. Here is the run and as can be seen it contains runs of "6666" and even "66666":

28032 --> 66600
28033 --> 66601
28034 --> 66602
28035 --> 66603
28036 --> 66604
28037 --> 66605
28038 --> 66606
28039 --> 66607
28040 --> 66610
28041 --> 66611
28042 --> 66612
28043 --> 66613
28044 --> 66614
28045 --> 66615
28046 --> 66616
28047 --> 66617
28048 --> 66620
28049 --> 66621
28050 --> 66622
28051 --> 66623
28052 --> 66624
28053 --> 66625
28054 --> 66626
28055 --> 66627
28056 --> 66630
28057 --> 66631
28058 --> 66632
28059 --> 66633
28060 --> 66634
28061 --> 66635
28062 --> 66636
28063 --> 66637
28064 --> 66640
28065 --> 66641
28066 --> 66642
28067 --> 66643
28068 --> 66644
28069 --> 66645
28070 --> 66646
28071 --> 66647
28072 --> 66650
28073 --> 66651
28074 --> 66652
28075 --> 66653
28076 --> 66654
28077 --> 66655
28078 --> 66656
28079 --> 66657
28080 --> 66660
28081 --> 66661
28082 --> 66662
28083 --> 66663
28084 --> 66664
28085 --> 66665
28086 --> 66666
28087 --> 66667
28088 --> 66670
28089 --> 66671
28090 --> 66672
28091 --> 66673
28092 --> 66674
28093 --> 66675
28094 --> 66676
28095 --> 66677

The end of the run, 28095, falls on March 5th 2026. 

Here is an infographic that summarises some of the information about hidden beast numbers.

Figure 1: generated by NotebookLM

Rhonda Numbers

Today I turned 25398 days old and I made Rhonda numbers the focus of my mathematical tweet of the day:

In base 8, 25398 is represented as 61466 and the product of these base 8 digits is 864. The sum of the prime factors (including multiplicity) of 25398 is 2+3*2+17+83 = 108 and 108 * 8 = 864. It's only the number itself that needs to be changed into a base other than 10. The prime factors and their sum all remain in base 10. So 25398 is a Rhonda number to base 8.

OEIS A100970 lists the initial (base 10) numbers that are Rhonda numbers in base 8 and they are:

1836, 6318, 6622, 10530, 14500, 14739, 17655, 18550, 25398, ...

Numbers Aplenty has this to say about the Rhonda numbers in general:

Rhonda numbers exist only in composite bases. Indeed, the product of the digits of a number in a prime base \(b\) cannot be divisible by \(b\), since every digit is smaller than \(b\). 

Kevin Brown has proved that there are infinite Rhonda numbers. 

The first base-10 Rhonda numbers are 1568, 2835, 4752, 5265, 5439, 5664, 5824, 5832, 8526, 12985, 15625, 15698, 19435, 25284. 

1000 is the smallest Rhonda number in two bases, namely 16 and 36, since: 

\(1000=2^3\cdot 5^3\), \(1000=(3,14,8)_{16}=(27,28)_{36}\) 

\(3\cdot14\cdot8 = 16\cdot(2\cdot3+3\cdot5)\) for base 16   

\(27\cdot28=36\cdot(2\cdot3+3\cdot5)\) for base 36. 

The first smallest Rhonda numbers with respect to one, two, ... , ten bases are 560, 1000, 10200, 5670, 63945, 158400, 322920, 140800, 1200420, 889200.

From MathWorld, the first few Rhonda numbers for small composite bases b are summarised in the following table:

	OEIS	Rhonda numbers
4	A100968	10206, 11935, 12150, 16031, 45030, 94185, ...
6	A100969	855, 1029, 3813, 5577, 7040, 7304, 15104, 19136, ...
8	A100970	1836, 6318, 6622, 10530, 14500, 14739, 17655, 18550, 25398, ...
9	A100973	15540, 21054, 25331, 44360, 44660, 44733, 47652, ...
10	A099542	1568, 2835, 4752, 5265, 5439, 5664, 5824, 5832, 8526, 12985, ...
12	A100971	560, 800, 3993, 4425, 4602, 4888, 7315, 8296, 9315, 11849, 12028, ...
14	A100972	11475, 18655, 20565, 29631, 31725, 45387, 58404, 58667, 59950, ...
15	A100974	2392, 2472, 11468, 15873, 17424, 18126, 19152, 20079, 24388, ...
16	A100975	1000, 1134, 6776, 15912, 19624, 20043, 20355, 23946, 26296, ...

Here is the SageMath code that I developed to generate the sequence in bases from 2 to 36 (with base 16 being used for the example):

b=16 #enter required base

L=[]

for x in range(2,27000):

    d=Integer(x).str(base=b)

    product=1

    for i in range(len(d)):

        product=product*int(d[i], base=b)

    P=list(factor(x))

    sum=0

    for p in P:

        sum+=p[0]*p[1]

    if product/sum ==b and product !=0:

        L.append(x)

print(L)

[1000, 1134, 6776, 15912, 19624, 20043, 20355, 23946, 26296] 

This algorithm will work for all bases from 2 up to 36 but there are no Rhonda numbers in prime bases, so there will be no output for these bases. 

Let's return to what was said in the Numbers Aplenty quote above, viz.:

The first smallest Rhonda numbers that can be written one base only, two bases, three bases etc. are: 

• one --> 560
• two --> 1000
• three --> 10200
• four --> 5670
• five --> 63945
• six --> 158400
• seven --> 322920
• eight --> 140800
• nine --> 1200420
• ten --> 889200

This site gives 140800 are an example. This is the smallest number that is a Rhonda number in eight different bases. See Figure 1.

Figure 1

Let's use base 198 to illustrate how this works. In base 198, 140800 can be represented as \( (3, 117, 22)_{198} \) and the product of these digits (3, 117 and 22) is 7722. Now 140800 factorises to 2^9 * 5^2 * 11 and the sum of these prime factors is 39 (remember we are working in base 10 here). If we multiply 39 by 198, we get 7722. Thus 140800 is a Rhonda number in base 198 and so it can be shown for the other bases.

on Thursday, April 1st 2021