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Rhonda Number


A positive integer n is called a base-b Rhonda number if the product of the base-b digits of n is equal to b times the sum of n's prime factors. These numbers were named by K. S. Brown after an acquaintance of his whose residence number 25662 satisfies this property. The etymology of the term is therefore similar to the Smith numbers.

25662 is a Rhonda number to base-10 since its prime factorization is

 25662=2·3·7·13·47
(1)

and the product of its base-10 digits satisfies

 2·5·6·6·2=720=10·(2+3+7+13+47).
(2)

The Rhonda numbers to base 10 are 1568, 2835, 4752, 5265, 5439, 5664, 5824, 5832, 8526, 12985, ... (OEIS A099542). The corresponding sums of prime factors are 24, 24, 28, 30, 54, 72, 32, 24, 48, 72, ... (OEIS A099543).

Rhonda numbers exist only for bases that are composite since there is no way for the product of integers less than a prime b to have b as a factor.

The first few Rhonda numbers for small composite bases b are summarized in the following table.

bOEISb-Rhonda numbers
4A10096810206, 11935, 12150, 16031, 45030, 94185, ...
6A100969855, 1029, 3813, 5577, 7040, 7304, 15104, 19136, ...
8A1009701836, 6318, 6622, 10530, 14500, 14739, 17655, 18550, 25398, ...
9A10097315540, 21054, 25331, 44360, 44660, 44733, 47652, ...
10A0995421568, 2835, 4752, 5265, 5439, 5664, 5824, 5832, 8526, 12985, ...
12A100971560, 800, 3993, 4425, 4602, 4888, 7315, 8296, 9315, 11849, 12028, ...
14A10097211475, 18655, 20565, 29631, 31725, 45387, 58404, 58667, 59950, ...
15A1009742392, 2472, 11468, 15873, 17424, 18126, 19152, 20079, 24388, ...
16A1009751000, 1134, 6776, 15912, 19624, 20043, 20355, 23946, 26296, ...

The smallest Rhonda number is 560, which is Rhonda to base 12. The integers that are Rhonda numbers to some base are n=560, 756, 800, 855, 1000, 1029, 1134, 1470, 1568, 1632, 1750, 1815, ... (OEIS A100987).

There exist integers that are Rhonda to more than one base. The smallest of these is 1000, which is Rhonda to bases 16 and 36, and the full sequence of these multiply Rhonda numbers begins 1000, 2940, 4200, 4212, 4725, 5670, 5824, ... (OEIS A100988).

That there are infinitely many Rhonda numbers can be seen from the following explicit construction. For any integer m>5, the number N=km(m+1)(2m+1)^2 is a Rhonda number to base B=2km(m+1), where k is any integer such that

 sopf(k)=m(m+1)-sopf(m)-sopf(m+1) 
 -2sopf(2m+1).
(3)

and sopf(k) denotes the sum of the prime factors of k. This equation is guaranteed to have at least one solution for k so long as m>5.

N is expressed in base B as

 N=d_0+d_1B=km(m+1)+2m(m+1)B
(4)

so the product of the base B digits of N is 2km^2(m+1)^2.

Because sopf is an additive function, we find that

 sopf(N)=sopf(k)+sopf(m)+sopf(m+1) 
 +2sopf(2m+1)=m(m+1)
(5)

where in the last step we have made use of (1). Therefore, B times the sum of the prime factors of N is equal to 2km^2(m+1)^2, which is equal to the product of the base B digits of N.

As an example, let us take m=6. Then we require that, from (1) above,

 sopf(k)=6·7-5-7-2·13=4
(6)

which is satisfied by k=4, and so N=4·6·7·13^2=28392 is a Rhonda number to base B=2·4·6·7=336.


See also

Smith Number

This entry contributed by Mark Hudson

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References

MathPages. "Smith Numbers and Rhonda Numbers." http://www.mathpages.com/home/kmath007.htm.MathPages. "Infinitely Many Rhondas." http://www.mathpages.com/home/kmath083.htm.Schneider, W. "Rhonda Numbers." http://www.wschnei.de/digit-related-numbers/rhonda-numbers.html.Sloane, N. J. A. Sequences A099542, A099543, A100968, A100969, A100970, A100971, A100972, A100973, A100974, A100975, A100987, and A100988 in "The On-Line Encyclopedia of Integer Sequences."

Referenced on Wolfram|Alpha

Rhonda Number

Cite this as:

Hudson, Mark. "Rhonda Number." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/RhondaNumber.html

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