[seqfan] Basic sequences in different arithmetics on N

[Vladimir Shevelev]

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