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%I #23 Aug 14 2022 02:47:18
%S 2,11,19,41,71,107,191,301,431,565,857,1133,1325,2657,5231,10457,
%T 19421,29567,54497,105527,211061,408431,802127,1600217,3200201,
%U 6393911,12783497,25566677,51095411,102190391,204347177,408693977,817302527,1634575487,3269107991
%N a(n) = b_f(n) where f is the 3-periodic sequence [-1,1,5] (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 f is a period 3 sequence with period [-1,1,5]. It appears [a(n),a(n)+2,a(n)+6] is a prime triple for n>=14 (a(n)>=2657).
%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) f(n)=[-1,1,5][(n+2)%3+1]
%o a=1; for(n=2, 1000000000, t=a; a=abs(a-gcd(a,n+f(n))); if(a==0, print1(n, ", ")))
%Y Cf. A186265.
%K nonn
%O 1,1
%A _Benoit Cloitre_, Feb 16 2011
%E More terms from _Jinyuan Wang_, Aug 14 2022
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