[seqfan] Re: Sloane's Sequence A023052

David Wilson davidwwilson at comcast.net
Mon Jun 15 03:03:55 CEST 2009

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.


  Yongwhan Lim

  P.S. I apologize if I used a wrong title; perhaps, Dr. might have been more suitable.


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