A186253 Indices of zeros of the sequence u(n)=abs(u(n-1)-gcd(u(n-1),n-1)), u(1)=1.

2, 5, 11, 23, 47, 79, 157, 313, 619, 1237, 2473, 4909, 9817, 19603, 39199, 78193, 156019, 311347, 622669, 1244149, 2487739, 4975111, 9950221, 19900399, 39800797, 79601461, 159202369, 318404629, 636788881, 1273577761, 2547155419, 5094310069, 10188620041

Offset: 1

Author

Benoit Cloitre, Feb 16 2011

Keywords

• nonn

Comments

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.

See A261301 for the sequence u relevant here (m=1). - M. F. Hasler, Aug 14 2015

A261301(a(n)-1) = 1; A261301(a(n)) = 0; A261301(a(n)+1) = a(n). - Reinhard Zumkeller, Sep 07 2015

Links

• Moritz Firsching, Table of n, a(n) for n = 1..315
• B. Cloitre, 10 conjectures in additive number theory, arXiv:1101.4274 [math.NT], 2011.
• M. F. Hasler, Rowland-Cloître type prime generating sequences, OEIS Wiki, August 2015.

Crossrefs

Cf. A106108.

Cf. A261301 - A261310; A186254 - A186263.

Programs

Haskell

a186253 n = a186253_list !! (n-1)
a186253_list = filter ((== 0) . a261301) [1..]
-- Reinhard Zumkeller, Sep 07 2015

Mathematica

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 *)
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 *)

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,",")))

PARI

next_a(last_a) = {
  local(A=last_a,B=last_a,C=2*last_a+1);
  while(A>0,
    D=divisors(C);
    k1=10*D[2];
    for(j=2,#D, d=D[j];k=((A+1-B+d)/2)%d;
      if(k==0,k=d); if(k<=k1,k1=k;d1=d));
    if(k1-1+d1==A,B=B+1);
    A = max(A-(k1-1)-d1,0);
    B = B + k1;
    C = C - (d1 - 1);
  );
  return(B);
}
a=2
for(n=1,99,print1(a,", ");a=next_a(a)) \\ Jan Büthe and Moritz Firsching, Aug 04 2015

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

Formula

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

Extensions

Definition clarified by M. F. Hasler, Aug 14 2015