[seqfan] On infinitude of A090918-A090919
shevelev at bgu.ac.il
Fri Nov 23 19:51:32 CET 2012
Consider the following interesting sequences.
A090918: Beginning with 3, least prime, greater than the previous term, such that the arithmetic mean of first n terms is a prime;
A090919: Primes arising as the arithmetic mean of first n terms of A090918.
The question of the infinitude of these sequences is reduced to unsolved
question, whether there are infinitly many primes in every "progression"
of the form a*prime(n)+b, gcd(a,b)=1. Indeed, I found the following recursive algorithm of parallel calculation of these sequences.
Let a(n)=A090919(n), b(n)=A090918(n). Then a(1)=b(1)=3 and, if for n>=2, we know a(n-1), b(n-1), then a(n) is the smallest prime x, such that
n*x-(n-1)*a(n-1) is prime more than b(n-1). Now, knowing a(n), we have
It is clear, that gcd (n, (n-1)*a(n-1))=1, so the infinitude of these sequences is reduced to above mentioned problem.
Are there other (maybe, probabilistic) arguments ?
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