Dan, You said: 1. (Aside to Neil: Maybe if room permits, each link to an associated sequence could have next to it a capsule description of itself?) Me: just hover the mouse over the A-number! 2. This suggests perhaps that the sequences of numerators and of denominators of the fractional parts might be interesting as well — as far as I could tell, these were not in OEIS. Me: Of course the first 43 numerators coincide with A003504, and grow very fast, so we essentially have them. As for the denominators, they begin (with offset 0) 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 43 - as you surmised, not yet in the OEIS - and if someone can send me a few more terms then I'll create an entry for them. Best regards Neil Neil J. A. Sloane, President, OEIS Foundation. 11 South Adelaide Avenue, Highland Park, NJ 08904, USA. Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ. Phone: 732 828 6098; home page: http://NeilSloane.com Email: njasloane@gmail.com On Tue, Nov 14, 2017 at 4:52 PM, Dan Asimov <dasimov@earthlink.net> wrote:
The entry http://oeis.org/A003504 for OEIS A003504 says:
----- a(0)..a(43) are integral, but from a(44) onwards every term is nonintegral -----
(a sentence that evidently took the collaboration of 3 authors to state correctly in full!).
This suggests perhaps that the sequences of numerators and of denominators of the fractional parts might be interesting as well — as far as I could tell, these were not in OEIS.
(Aside to Neil: Maybe if room permits, each link to an associated sequence could have next to it a capsule description of itself?)
Question: --------- Is there some way to formalize the kind of property that a(n) exhibits and to prove that a(n) (or some other sequence?) is the "simplest" one with that property?
—Dan
Tom Rockicki wrote: -----
I'm curious to find out what the fractional part of the number will turn out to be. I think it'll be some integer divided by 43.
As shown by Guy in the paper (which is scanned with annotations here: http://oeis.org/A005165/a005165.pdf) you'll see that the fractional part is simply 24/43 (see page 709, near his written A3504.)
On Tue, Nov 14, 2017 at 7:47 AM, Hans Havermann <gladhobo@bell.net> wrote:
Yesterday, Cliff Pickover's twitter feed presented a bit from Pickover's 2005 "A Passion for Mathematics" which references Guy's 1994 "Unsolved Problems in Number Theory" (2nd ed.) E15, a recursion of Göbel, wherein is stated that x(43) of the sequence is not an integer. The sequence is A003504:
There's a different offset in the OEIS version, so A003504(44) is now the first one that is not an integer. Pickover in his book felt the need to add something to the problem so, noting that A003504(44) = 5.4093*10^178485291567, he stated that this number "is so large that humanity will *never* be able to compute all of its digits".
I had a go on my four-year-old Mac Pro with 64 GB RAM and was only able to compute A003504(42) with its 44621322894 decimal digits. That suggests that when the next iteration of the Mac Pro, with 256 GB RAM, comes out in 2018 it should be able to calculate the number. But I know that there are personal computer setups out there right now that enjoy 256 GB RAM, so I emailed Cliff with a "never is now". :)
I'm curious to find out what the fractional part of the number will turn out to be. I think it'll be some integer divided by 43.
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