mersenneforum.org Gcd of consecutive factorial sums
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2019-11-27, 02:18   #12
LaurV
Romulan Interpreter

Jun 2011
Thailand

19·467 Posts

Quote:
 Originally Posted by R.D. Silverman As John Selfridge said: We know log log n goes to infinity. But no one has ever observed it doing so.
This is brilliant! Let me have it as a motto on my skype for few days!

(I say that every time when I change my motto, then I forget about it or don't find anything better, and the motto stays for some weeks...)

 2019-11-27, 08:06 #13 axn     Jun 2003 22×32×131 Posts Searched this up to 18million. No hits. Not proceeding any further.
2019-11-27, 16:25   #14
carpetpool

"Sam"
Nov 2016

22×79 Posts

Quote:
 Originally Posted by Uncwilly Which thread. A mod can move the post.
https://www.mersenneforum.org/showpo...10&postcount=8

2019-11-27, 17:02   #15
R. Gerbicz

"Robert Gerbicz"
Oct 2005
Hungary

17·83 Posts

Quote:
 Originally Posted by axn Searched this up to 18million. No hits. Not proceeding any further.
Not interested in the actual computation, but you can do it in polynomial time per prime if you want all residues for p<N, similarly to the computation of Wilson (prime) residues.

Last fiddled with by R. Gerbicz on 2019-11-27 at 17:03 Reason: typo, grammar

 2020-06-06, 19:25 #16 Citrix     Jun 2003 62716 Posts I have been thinking of a generalized version of the above problem We can define S(n)=S(n-1)+n*(S(n-1)-S(n-2)) For every value of S(0) and S(1) we can get a different sequence (where S(1)>S(0)) Conjecture:- For every unique sequence there a prime p such that S(p+x) is always divisible by p (x being a positive natural number) Some thoughts If a prime p divides the sequence for a particular S(0) and S(1) Then prime p also divides all sequences where S'(0)=k*S(0) and S'(1)=k*S(1) (k is a natural number) So we only need to look at the sequence with primitive solutions where S(0)=1 and S(1)>1 S(0)=0 and S(1)>0 is a special case which leads to multiple of the sequence posted in the initial post of this thread. I searched S(0)=1 and S(1)<10000 for all primes p<1,000,000. There are 706 sequences left. Any further thoughts? If the above conjecture is false - what it the density of remaining sequences?

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