A186253 - OEIS

%I #39 Oct 02 2022 14:51:56

%S 2,5,11,23,47,79,157,313,619,1237,2473,4909,9817,19603,39199,78193,

%T 156019,311347,622669,1244149,2487739,4975111,9950221,19900399,

%U 39800797,79601461,159202369,318404629,636788881,1273577761,2547155419,5094310069,10188620041

%N Indices of zeros of the sequence u(n)=abs(u(n-1)-gcd(u(n-1),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=1 it appears a(n) is prime for n>=1.

%C See A261301 for the sequence u relevant here (m=1). - _M. F. Hasler_, Aug 14 2015

%C A261301(a(n)-1) = 1; A261301(a(n)) = 0; A261301(a(n)+1) = a(n). - _Reinhard Zumkeller_, Sep 07 2015

%H Moritz Firsching, <a href="/A186253/b186253.txt">Table of n, a(n) for n = 1..315</a>

%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 Conjecture: a(n) is asymptotic to c*2^n with c = 1.1861...

%t a = m = 1; Reap[For[n = 2, n <= 10^7, n++, a = Abs[a - GCD[a, m*n - 1]]; If[a == 0, Print[m*n + m - 1]; Sow[m*n + m - 1]]]][[2, 1]] (* _Jean-François Alcover_, Feb 05 2019, from PARI *)

%t nxt[{n_,a_}]:={n+1,Abs[a-GCD[a,n]]}; Position[NestList[nxt,{1,1},13*10^5][[All,2]],0]// Flatten (* The program generates the first 20 terms of the sequence. *) (* _Harvey P. Dale_, Oct 02 2022 *)

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

%o (PARI)

%o next_a(last_a) = {

%o   local(A=last_a,B=last_a,C=2*last_a+1);

%o   while(A>0,

%o     D=divisors(C);

%o     k1=10*D[2];

%o     for(j=2,#D, d=D[j];k=((A+1-B+d)/2)%d;

%o       if(k==0,k=d); if(k<=k1,k1=k;d1=d));

%o     if(k1-1+d1==A,B=B+1);

%o     A = max(A-(k1-1)-d1,0);

%o     B = B + k1;

%o     C = C - (d1 - 1);

%o   );

%o   return(B);

%o }

%o a=2

%o for(n=1,99,print1(a,", ");a=next_a(a)) \\ _Jan Büthe_ and _Moritz Firsching_, Aug 04 2015

%o (PARI) m=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

%o (Haskell)

%o a186253 n = a186253_list !! (n-1)

%o a186253_list = filter ((== 0) . a261301) [1..]

%o -- _Reinhard Zumkeller_, Sep 07 2015

%Y Cf. A106108.

%Y Cf. A261301 - A261310; A186254 - A186263.

%K nonn

%O 1,1

%A _Benoit Cloitre_, Feb 16 2011

%E Definition clarified by _M. F. Hasler_, Aug 14 2015