# [seqfan] Re: Sloane's Sequence A023052

Robert G. Wilson, v rgwv at rgwv.com
Wed Jun 17 07:08:19 CEST 2009

```Dear Sir,

See http://mathworld.wolfram.com/NarcissisticNumber.html

Sincerely yours, Bob.

----- Original Message -----
From: "David Wilson" <davidwwilson at comcast.net>
To: "Yongwhan Lim" <yongwhan at stanford.edu>
Cc: "Sequence Fans" <seqfan at seqfan.eu>
Sent: Sunday, June 14, 2009 9:03 PM
Subject: [seqfan] Re: Sloane's Sequence A023052

> The short answer is no, we do not know whether A023052 is finite.
>
> For "numbers that are the sum of the kth powers of their digits" for some fixed
> k, or "d-digit numbers that are the sum of the dth power of their digits"
> (narcissistic numbers), you can prove that the numbers in question have a
> bounded number of digits, so that the sequences must be finite. But for
> powerful(3) numbers, which are "numbers that are the sum of the kth power of
> their digits for some k" (A023052), there is no such argument, there is no
> bound on the number of digits. So we cannot prove A023052 finite by the same
> sort of argument, on the other hand, to prove the A023052 infinite, the most
> likely proof would be to exhibit an infinite subset, but this has never been
> done and seems unlikely. So it is not known if A023052 is finite or infinite,
> and we probably won't for a very long time if ever.
>
> At one time, Gusev had a list of all elements of A023052 up to 10^50 online. At
> that time, the OEIS did not include b-files, or I would probably have copied
> this list to the OEIS. By the time b-files were establish, Gusev's list had
> disappeared and I have not been able to find it.
>
> ----- Original Message -----
>  From: Yongwhan Lim
>  To: davidwwilson at comcast.net ; GGN at rm.yaroslavl.ru
>  Sent: Sunday, June 14, 2009 4:43 PM
>  Subject: Sloane's Sequence A023052
>
>
>  Dear Mr. Wilson and Mr. Gusev:
>
>  I am currently an undergraduate student majoring in Mathematics and Computer
> Science. A day ago, I faced a question concerning the special case of powerful
> numbers (where the exponent is 3) in Project Euler. As I was more interesting
> in the more general question (concerning any exponent), I ran a brute-force
> computation in C++. Facing difficulty with computation, I began searching
> through the web. I found out that this set of number is actually called a
> powerful number, whence I found the sequence in the Sloane's website.
>
>  I have just a quick question about Sloane's Sequence A023052: Powerful
> Numbers.
>
>  I would like to know what kind of results are currently known about these set
> of numbers: the most important question that I would like to settle is whether
> finiteness of this set of numbers proven yet (analogous to 88 narcissistic
> numbers). I tried to search for some papers concerning the number in
> MathSciNet, but I am facing a difficulty as powerful number has a multiple
> definition, the most common one being the one studied by Erdos and Szekeres.
>
>  Please let me know, other than what appears in the webpage
> (http://www.research.att.com/~njas/sequences/A023052), what else is currently
>
>  I am looking forward to hearing from you.
>
>  Thank you very much in advance.
>
>  Sincerely,
>
>  Yongwhan Lim
>
>  P.S. I apologize if I used a wrong title; perhaps, Dr. might have been more
> suitable.
>
>
>
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