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%I #35 May 13 2023 13:46:05
%S 14,21,22,23,24,25,26,39,40,45,48,49,56,57,58,59,60,61,62,63,64,65,66,
%T 67,68,69,70,71,72,73,74,111,112,113,114,115,116,117,118,119,126,127,
%U 128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143
%N Cousin prime recurrence sequence: a(1) = 14, and for n > 1, a(n) = a(n-1) + gcd(n+5, a(n-1)), if n is even, otherwise a(n) = a(n-1) + gcd(n+1, a(n-1)).
%C 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.)
%C 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.)
%C 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)).
%H T. D. Noe, <a href="/A213536/b213536.txt">Table of n, a(n) for n = 1..1273</a>
%H G. H. Hardy and J. E. Littlewood, <a href="https://dx.doi.org/10.1007/BF02403921">Some problems of 'Partitio numerorum'; III: on the expression of a number as a sum of primes</a>, Acta Mathematica, Vol. 44, pp. 1-70, 1923.
%H 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 Pascal Sebah and Xavier Gourdon, <a href="http://numbers.computation.free.fr/Constants/Primes/twin.html">Introduction to twin primes and Brun's constant</a>.
%H Vladimir Shevelev, <a href="https://arxiv.org/abs/0910.4676">A new generator of primes based on the Rowland idea</a>, arXiv:0910.4676 [math.NT], 2009.
%H Vladimir Shevelev, <a href="https://arxiv.org/abs/0911.5478">Three theorems on twin primes</a>, arXiv:0911.5478 [math.NT], 2009-2010.
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Dickson's_conjecture">Dickson's conjecture</a>.
%p A213536 := proc(n)
%p option remember;
%p if n = 1 then
%p 14;
%p elif type(n,'even') then
%p procname(n-1)+gcd(n+5,procname(n-1)) ;
%p else
%p procname(n-1)+gcd(n+1,procname(n-1)) ;
%p end if;
%p end proc: # _R. J. Mathar_, Jul 20 2012
%t 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 *)
%Y Cf. A213537 (record differences).
%K nonn
%O 1,1
%A _Joseph Benstock_, Jun 27 2012
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