A186257 - OEIS

%I #12 Jan 12 2016 10:36:51

%S 14,89,479,2879,17099,99839,599009,3592859,21557099,129336149,

%T 775914479,4655486369,27932918219,167597509319,1005582321329,

%U 6033492323549,36200953941059,217205705087639,1303234230378959,7819405361540219

%N a(n) = 5*b_5(n)+4, where b_5 lists the indices of zeros of the sequence A261305: u(n) = abs(u(n-1)-gcd(u(n-1),5*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 k large enough m*b_m(k)+m-1 is a prime number. Here for m=5 it appears a(n) is prime for n>=2.

%C See A261305 for the sequence u relevant here (m=5). - _M. F. Hasler_, Aug 14 2015

%H B. Cloitre, <a href="http://arxiv.org/abs/1101.4274">10 conjectures in additive number theory</a>, preprint arxiv:2011.4274 (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 We conjecture that a(n) is asymptotic to c*6^n with c>0.

%F See the wiki link for a sketch of a proof of this conjecture. More precisely we find c = 1.15917467761... - _M. F. Hasler_, Aug 22 2015

%o (PARI) a=1; m=5; for(n=2, 1e7, a=abs(a-gcd(a, m*n-1)); if(a==0, print1(m*n+m-1, ", ")))

%o (PARI) m=5; a=k=1; for(n=1, 25, 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.

%Y Cf. A261301 - A261310; A186253 - A186263.

%K nonn

%O 1,1

%A _Benoit Cloitre_, Feb 16 2011

%E Edited by _M. F. Hasler_, Aug 14 2015

%E More terms from _M. F. Hasler_, Aug 22 2015