OFFSET
1,1
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
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.
FORMULA
Conjecture: a(n) is asymptotic to c*2^n with c = 1.1861...
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 *)
PROG
(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
(Haskell)
a186253 n = a186253_list !! (n-1)
a186253_list = filter ((== 0) . a261301) [1..]
-- Reinhard Zumkeller, Sep 07 2015
CROSSREFS
KEYWORD
nonn
AUTHOR
Benoit Cloitre, Feb 16 2011
EXTENSIONS
Definition clarified by M. F. Hasler, Aug 14 2015
STATUS
approved