|
| |
|
|
A167168
|
|
Sequence of prime gaps which characterize Rowland sequences of prime-generating recurrences.
|
|
11
|
|
|
|
3, 7, 17, 19, 31, 43, 53, 67, 71, 79, 97, 103, 109, 113, 127, 137, 151, 163, 173, 181, 191, 197, 199, 211, 229, 239, 241, 251, 257, 269, 271, 283, 293, 317, 331, 337, 349, 367, 373
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
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
|
|
|
|
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
|
|
|
|
EXTENSIONS
|
Edited, a(1) set to 3, 37 replaced by 31, and extended beyond 53 by R. J. Mathar, Dec 17 2009
|
|
|
STATUS
|
approved
|
| |
|
|
