[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 
> known about the sequence.
>  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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