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

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

Offset: 1

Author

Vladimir Shevelev, Oct 29 2009

Keywords

• nonn

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.

Examples

			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,..}.
		

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.

Crossrefs

Cf. A106108, A167053, A116533, A163963, A084662, A084663, A134162.

Extensions

Edited, a(1) set to 3, 37 replaced by 31, and extended beyond 53 by R. J. Mathar, Dec 17 2009