cp's OEIS Frontend

A167168 Sequence of prime gaps which characterize Rowland sequences of prime-generating recurrences.

%I A167168 #7 May 05 2015 17:16:53
%S A167168 3,7,17,19,31,43,53,67,71,79,97,103,109,113,127,137,151,163,173,181,
%T A167168 191,197,199,211,229,239,241,251,257,269,271,283,293,317,331,337,349,
%U A167168 367,373
%N A167168 Sequence of prime gaps which characterize Rowland sequences of prime-generating recurrences.
%C A167168 Consider the Rowland sequences with recurrence N(n)= N(n-1)+gcd(n,N(n-1)).
%C A167168 For some of these, like the prototypical A106108, the first differences N(n)-N(n-1) are always 1 or primes.
%C A167168 If for some position p (a prime) N(p-1)=2*p, then the arXiv preprint shows that N is indeed in that class of prime-generating sequences.
%C A167168 Since then N(p)=N(p-1)+p, the prime p characterizes at the same time the gap (first difference) and location in the sequence.
%C A167168 In the same sequence at some larger value of p, we may again have N(p-1)=2*p. In these cases, we put all these p's satisfying that equation into a generator class.
%C A167168 For each of the generator classes, the OEIS sequence shows the smallest member (prime) in that class. So this is a trace of how many essentially different sequences with this N(p-1)=2*p property exist.
%H A167168 E. S. Rowland, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL11/Rowland/rowland21.html">A natural prime-generating recurrence</a>, Journal of Integer Sequences, Vol. 11 (2008), Article 08.2.8
%H A167168 V. Shevelev, <a href="http://arXiv.org/abs/0910.4676">A new generator of primes based on the Rowland idea</a>, arXiv:0910.4676 [math.NT], 2009.
%e A167168 We put a(1)=3 since the N-sequence 4, 6, 9, 10, 15, 18, 19, 20.. = A084662 (essentially the same as A106108) has a first difference of p=3 at position p-1=2, N(2)=2*3.
%e A167168 It has a first difference of p=5 at p-1=4, a first difference of p=11 at p=10, so we put {3,5,11,23,..} into that class. This leaves p=7=a(2) as the lowest prime to be covered by the next class. This is first realized by N = 8, 10, 11, 12, 13, 14, 21, 22, 23, 24, 25, 26, 39.. = A084663. Here N(12)=2*13, so p=13 is in the same class as p=7, namely {7,13,29,59,131,..}. This leaves p=17=a(3) to be the smallest member in a new class, namely {17,41,83,167,..}.
%Y A167168 Cf. A106108, A167053, A116533, A163963, A084662, A084663, A134162.
%K A167168 nonn
%O A167168 1,1
%A A167168 _Vladimir Shevelev_, Oct 29 2009
%E A167168 Edited, a(1) set to 3, 37 replaced by 31, and extended beyond 53 by _R. J. Mathar_, Dec 17 2009