A339458 - OEIS

%I #11 Jan 11 2021 23:35:42

%S 1,1,1,63,7,3,2,2,1,1,1,250,2,1,2,1,2,3,1,4,1,1,3,1,1,2,1,1,1,2,1,2,2,

%T 1,1,6,7,1,1,1,6,1,1,9,9,2,1,6,2,5,1,25,1,1,1,2,18,1,3,5,1,1,5,1,3,1,

%U 1,4,1,1,3,2,2,3,40,2,3,8,2,2,25,1,5,2,1,1,3,2,2,1,10,1,1,2,1,2,1,1,2,1,3,2,420,2,2,1

%N Continued fraction expansion of the smallest constant d such that the numbers floor(2^(n^d)) are distinct primes for all n >= 1.

%e 1+1/(1+1/(1+1/(63+1/(7+1/(3+1/(2+1/(2+1/(1+1/(1+1/(1+1/(250] = 22739482/15120055 = 1.503928524069522...

%e The constant is equal to d=1.503928524069520633527689067897583199190738849581138429002999...

%o (PARI) A339458(n=63, prec=200)={

%o   my(curprec=default(realprecision));

%o   default(realprecision, max(prec,curprec));

%o   my(a=List([2]), d=1.0, c=2.0, b, p, ok, smpr(b)=my(p=b); while(!isprime(p), p=nextprime(p+1)); return(p); );

%o   for(j=1, n-1,

%o     b=floor(c^(j^d));

%o     until(ok,

%o       p=smpr(b);

%o       ok = 1;

%o       listput(a,p,j);

%o       if(p!=b,

%o          d=log(log(p)/log(c))/log(j);

%o          for(k=1,j-2,

%o              b=floor(c^(k^d));

%o              if(b!=a[k],

%o                 ok=0;

%o                 j=k;

%o                 break;

%o                );

%o             );

%o         );

%o     );

%o   );

%o   default(realprecision, curprec);

%o   return(contfrac(d));

%o } \\ _François Marques_, Dec 08 2020

%Y Cf. A339459, A339457, A338613, A338837, A338850.

%K nonn,cofr

%O 1,4

%A _Bernard Montaron_, Dec 06 2020