OFFSET
1,1
COMMENTS
Conjecture: Record differences a(n) - a(n-1) (A213537) are a strict subset of the smaller of cousin primes (A023200). (Cousin primes differ by 4.)
Conjecture: Record differences are an infinite sequence. It is widely believed there are infinitely many cousin primes. (Similarly, by Dickson's conjecture and the second Hardy-Littlewood conjecture, there are infinitely many pairs of (not necessarily consecutive) primes (p, p+2k) for each natural number k.)
Conjecture: The following pattern makes sequences for every (necessarily even) difference (slight change for 2). For difference d, p is first prime > d that is the smaller of a prime pair (p, p+d). a(1) = 2p and a(n) = gcd(n+p-2, a(n-1)) for even n, otherwise gcd(n+p-2-d, a(n-1)).
LINKS
T. D. Noe, Table of n, a(n) for n = 1..1273
G. H. Hardy and J. E. Littlewood, Some problems of 'Partitio numerorum'; III: on the expression of a number as a sum of primes, Acta Mathematica, Vol. 44, pp. 1-70, 1923.
E. S. Rowland, A natural prime-generating recurrence, Journal of Integer Sequences, Vol. 11 (2008), Article 08.2.8.
Pascal Sebah and Xavier Gourdon, Introduction to twin primes and Brun's constant.
Vladimir Shevelev, A new generator of primes based on the Rowland idea, arXiv:0910.4676 [math.NT], 2009.
Vladimir Shevelev, Three theorems on twin primes, arXiv:0911.5478 [math.NT], 2009-2010.
Wikipedia, Dickson's conjecture.
MAPLE
A213536 := proc(n)
option remember;
if n = 1 then
14;
elif type(n, 'even') then
procname(n-1)+gcd(n+5, procname(n-1)) ;
else
procname(n-1)+gcd(n+1, procname(n-1)) ;
end if;
end proc: # R. J. Mathar, Jul 20 2012
MATHEMATICA
nxt[{n_, a_}]:={n+1, If[OddQ[n], a+GCD[a, n+6], a+GCD[a, n+2]]}; Transpose[ NestList[nxt, {1, 14}, 60]][[2]] (* Harvey P. Dale, Jun 22 2013 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Joseph Benstock, Jun 27 2012
STATUS
approved