OFFSET
1,1
COMMENTS
Consider the Rowland sequences with recurrence N(n)= N(n-1)+gcd(n,N(n-1)).
For some of these, like the prototypical A106108, the first differences N(n)-N(n-1) are always 1 or primes.
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.
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.
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.
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.
LINKS
E. S. Rowland, A natural prime-generating recurrence, Journal of Integer Sequences, Vol. 11 (2008), Article 08.2.8
V. Shevelev, A new generator of primes based on the Rowland idea, arXiv:0910.4676 [math.NT], 2009.
EXAMPLE
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.
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,..}.
CROSSREFS
KEYWORD
nonn
AUTHOR
Vladimir Shevelev, Oct 29 2009
EXTENSIONS
Edited, a(1) set to 3, 37 replaced by 31, and extended beyond 53 by R. J. Mathar, Dec 17 2009
STATUS
approved