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

Numbers Aplenty

Today I just stumbled upon a new site that lists information about natural numbers. It's called Numbers Aplenty.

The categories under which numbers are classified looks like this:

So for instance, today I am 25284 days old and entering that number into the search box yields:

• 25284 has 36 divisors (see below), whose sum is σ = 70224. Its totient is φ = 7056. 

• The previous prime is 25261. The next prime is 25301. The reversal of 25284 is 48252 

• Adding to 25284 its reverse (48252), we get a triangular number (73536 = T383). 

• It is a Harshad number since it is a multiple of its sum of digits (21). 

• 25284 is a Rhonda number in base 10. 

• Its product of digits (640) is a multiple of the sum of its prime factors (64). 

• It is a nialpdrome in base 14. 

• It is a self number, because there is not a number n which added to its sum of digits gives 25284. 

• It is an unprimeable number. 

• It is a polite number, since it can be written in 11 ways as a sum of consecutive naturals, for example, 567 + ... + 609. 

• 225284 is an apocalyptic number. 

• It is an amenable number. 

• It is a practical number, because each smaller number is the sum of distinct divisors of 25284, and also a Zumkeller number, because its divisors can be partitioned in two sets with the same sum (35112). 

• 25284 is an abundant number, since it is smaller than the sum of its proper divisors (44940). 

• It is a pseudoperfect number, because it is the sum of a subset of its proper divisors. 

• 25284 is a wasteful number, since it uses less digits than its factorization. 

• 25284 is an evil number, because the sum of its binary digits is even. 

• The sum of its prime factors is 64 (or 55 counting only the distinct ones). 

• The product of its digits is 640, while the sum is 21. 

• The square root of 25284 is about 159.0094336824. The cubic root of 25284 is about 29.3504835430. 

• The spelling of 25284 in words is "twenty-five thousand, two hundred eighty-four".

Many of the terms mentioned above I'd never heard before. The site will be a useful adjunct to the Online Encyclopedia of Integer Sequences or OEIS.