A186255 a(n) = 3*b_3(n)+2, where b_3 lists the zeros of the sequence A261303: u(n+1)=abs(u(n)-gcd(u(n),3*n+2)), u(1)=1.

8, 17, 71, 269, 1013, 4007, 15923, 63521, 253949, 1014317, 4056893, 16225589, 64902359, 259609439, 1038437759, 4153750883, 16614561281, 66458241569, 265832966279, 1063331407109, 4253325628439, 17013302513759, 68053207705097, 272212800371669, 1088851201483883

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=3 it appears a(n) is prime for n>=2.

See A261303 for the sequence u relevant here (m=3). - M. F. Hasler, Aug 14 2015

Links

• B. Cloitre, 10 conjectures in additive number theory, preprint arxiv:2011.4274
• M. F. Hasler, Rowland-Cloître type prime generating sequences, OEIS Wiki, August 2015.

Crossrefs

Cf. A106108.

Cf. A261301 - A261310; A186253 - A186263.

Programs

PARI

a=1; m=3; for(n=2, 10^7, a=abs(a-gcd(a, m*n-1)); if(a==0, print1(m*n+m-1, ", ")))

PARI

m=3; 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

Formula

We conjecture that a(n) is asymptotic to c*4^n with c=0.96...

See the wiki link for a sketch of a proof that this is true. We can give more decimals of c = 0.967094... - M. F. Hasler, Aug 22 2015

Extensions

More terms from M. F. Hasler, Aug 22 2015