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%I #20 Jan 12 2016 10:51:56
%S 5,17,53,149,449,1289,3761,11261,33773,101117,302681,907757,2723069,
%T 8169137,24506597,73519793,220559369,661677761,1985001917,5955003077,
%U 17865008333,53595020201,160785060361,482355180761,1447065541373,4341196624109,13023589872329
%N a(n) = 2*b(n)+1, where b(n) lists the zeros of the sequence u(n)=abs(u(n-1)-gcd(u(n-1),2*n-1)), u(1)=1.
%C For any fixed integer m>=1 define u(1)=1 and u(n)=abs(u(n-1)-gcd(u(n-1),m*n-1)). Then (b_m(k))_{k>=1} is the sequence of integers such that u(b_m(k))=0 and we conjecture that for all k large enough, m*b_m(k)+m-1 is a prime number. Here for m=2 it appears a(n) is prime for n>=1.
%C See A261302 for the sequence u relevant here (m=2). - _M. F. Hasler_, Aug 14 2015
%H B. Cloitre, <a href="http://arxiv.org/abs/1101.4274">10 conjectures in additive number theory</a>, arXiv:1101.4274 [math.NT], 2011.
%H M. F. Hasler, <a href="https://oeis.org/wiki/User:M._F._Hasler/Work_in_progress/Rowland-Cloitre_type_prime_generating_sequences">Rowland-Cloître type prime generating sequences</a>, OEIS Wiki, August 2015.
%F a(n+1) <= 3*a(n)+2 for all n.- See the wiki link for a sketch of a proof that a(n) ~ c*3^n with c = 1.7078779... - _M. F. Hasler_, Aug 22 2015
%o (PARI) a=1; m=2; for(n=2, 9e9, if(!a=abs(a-gcd(a, m*n-1)), print1(m*n+m-1, ", ")))
%o (PARI) m=2; a=k=1; for(n=1, 30, while( a>D=vecmin(apply(p->a%p, factor(N=m*(k+a)+m-1)[, 1])), a-=D+gcd(a-D, N); k+=1+D); k+=a+1; print1(a=N, ", ")) \\ _M. F. Hasler_, Aug 22 2015
%Y Cf. A106108, A186253 - A186263.
%Y Cf. A261301 - A261310.
%K nonn
%O 1,1
%A _Benoit Cloitre_, Feb 16 2011
%E More terms from _M. F. Hasler_, Aug 22 2015