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Old 2019-11-27, 02:18   #12
Romulan Interpreter
LaurV's Avatar
Jun 2011

19·467 Posts

Originally Posted by R.D. Silverman View Post
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...)
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Old 2019-11-27, 08:06   #13
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Jun 2003

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Searched this up to 18million. No hits. Not proceeding any further.
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Old 2019-11-27, 16:25   #14
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Nov 2016

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Originally Posted by Uncwilly View Post
Which thread. A mod can move the post.
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Old 2019-11-27, 17:02   #15
R. Gerbicz
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"Robert Gerbicz"
Oct 2005

17·83 Posts

Originally Posted by axn View Post
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
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Old 2020-06-06, 19:25   #16
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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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