[seqfan] Re: Basic sequences in different arithmetics on N

[Vladimir Shevelev]

Unfortunately,  in the second part of my message I missed a key word: instead of  [there exists the maximal possible "prime"], it should be [there exists the maximal possible exponent of a "prime"].
 
Sorry,
Vladimir


----- Original Message -----
From: Vladimir Shevelev <shevelev at bgu.ac.il>
Date: Monday, September 9, 2013 0:18
Subject: [seqfan] Basic sequences in different arithmetics on N
To: Sequence Fanatics Discussion list <seqfan at list.seqfan.eu>

> 
> Dear SeqFans,
>  
> Let us call an increasing positive integer sequence B={b_n} a 
> basic sequence for an arithmetic A on the set  N of all 
> positive integers, if the following conditions satisfy 1) B not 
> contains 1;  2) there exists a set E of positive integers 
> such that every integer >1 is a  product of powers of B-
> numbers with exponents from E; 3) the latter representation is 
> unique. Then A is called A(B,E)-arithmetic, and  B is 
> called  A(B,E)-primes. What is about the usual arithmetic? 
> Here B=P which is the set of all primes and E=N. In "Fermi-Dirac 
> -arithmetic" B=A050376 and E={1}. If for a fixed k>=1 to 
> consider as B the increasing-ordered sequence of numbers of the 
> form p^((k+1)^m) where p is prime, m >= 0, then it is iasy to 
> prove that B is a basic sequence with E={1,2,...,k}. For 
> example, A186285  is basic sequence with E={1,2}. Thus, if 
> k tends to infinity, we obtain
> the usual arithmetic. It is interesting that for more 
> complicated E's there exist infinitely many other basic sequences.
> It is interesting also that usual arithemetic and Fermi-Dirac 
> one has a common peculiarity: in both of them there is no a sense
> to say on numbers such that in their product-representation 
> there exists the maximal possible "prime". In usual arithmetic 
> there are no such numbers, while in  Fermi-Dirac one there 
> are no others. In other arithmetics such sequences there exist. 
> For example, in case of basic sequence A186285, such a sequence 
> is A177880.
>  
> Best regards,
> Vladimir
> 
>  Shevelev Vladimir‎
> 
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 Shevelev Vladimir‎