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%I #28 May 20 2024 10:29:41
%S 2,5,11,23,41,61,107,197,311,617,1229,2381,4649,8861,17027,33809,
%T 67409,134681,267719,535349,1069217,2138399,4275641,8545697,17091377,
%U 34182749,68365469,136730639,273461159,546917141,1093813727,2187610991,4375221077,8750432231
%N a(n) = b_f(n) where f is the 2-periodic sequence f(k) = (-1)^k (see comments).
%C Let u(1)=1 and u(n)=abs(u(n-1)-gcd(u(n-1),n+f(n))) where f(n) is a periodic sequence with period [f(1),f(2),...,f(beta)]. Then (b_f(k))_{k>=1} is the sequence of integers such that u(b_f(k))=0. We conjecture that for k large enough b_f(k)+1+f(i) is simultaneously prime for i=1,2,...,beta. Here for f(k)=(-1)^k it appears a(n) and a(n)+2 are twin primes for n>=7. If we start with u(1) large enough (such as with u(1)=71) the sequence will produce only twin primes.
%H Benoit Cloitre, <a href="http://arxiv.org/abs/1101.4274">10 conjectures in additive number theory</a>, arXiv:1101.4274 [math.NT], 2011.
%F Conjecture: a(n) is asymptotic to c*2^n with c>0.
%o (PARI) a=1; for(n=2, 10^9, a=abs(a-gcd(a,n+(-1)^n)); if(a==0, print1(n, ", ")))
%Y Cf. A186267.
%K nonn
%O 1,1
%A _Benoit Cloitre_, Feb 16 2011
%E More terms from _Jinyuan Wang_, Jan 09 2021