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 baseL=[]for x in range(2,27000):d=Integer(x).str(base=b)product=1for i in range(len(d)):product=product*int(d[i], base=b)P=list(factor(x))sum=0for 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
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