[seqfan] arithmetic index of a polynomial

Vladimir Shevelev shevelev at bgu.ac.il
Sun Mar 17 12:28:16 CET 2013


 
Dear SeqFans,
 
Consider sequence A213671: the odd part of n^2 - n + 2.  Calculating its values by handy, I was surprised that its composite terms were multiple of previous prime terms. Thus up to the moment when this phenomenon will break, we can obtain prime terms of the sequence using Eratosthenes-like sieve and thus numbers of the form 2^k*p among polynomial's values, where p is prime. Continuing this sequence, Peter Moses found that the first composite term for which the rule breaks is a(245) = 29891 = 71*421 - neither of 71, 421 are terms before a(245). In connection with this experiment I decided to introduce an "arithmetic index" of a polynomial P(n): it is the smallest m, such that in the sequence "odd part of P(n)" P(m) is composite and not multiple of any previous prime term. I asked Peter to find indices of polynomials of some series. I give only one example of series of Peter's calculations:n^2 + k, n>=1. I am writing ind(k): ind(1)=41, ind(2)=24, ind(3)=88, ind(4)=157, ind(5)=15, ind(6)=9, ind(7)=302, etc.
It is natural to pose an important question: is the set of indices of polynomials, say, of order 2 unbounded?
The case of unboundedness would be very desired especially in its possible constructive aspect.
 
Best regards,
Vladimir

 Shevelev Vladimir‎



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