cp's OEIS Frontend

A339458 Continued fraction expansion of the smallest constant d such that the numbers floor(2^(n^d)) are distinct primes for all n >= 1.

%I A339458 #11 Jan 11 2021 23:35:42
%S A339458 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 A339458 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 A339458 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 A339458 Continued fraction expansion of the smallest constant d such that the numbers floor(2^(n^d)) are distinct primes for all n >= 1.
%e A339458 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 A339458 The constant is equal to d=1.503928524069520633527689067897583199190738849581138429002999...
%o A339458 (PARI) A339458(n=63, prec=200)={
%o A339458   my(curprec=default(realprecision));
%o A339458   default(realprecision, max(prec,curprec));
%o A339458   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 A339458   for(j=1, n-1,
%o A339458     b=floor(c^(j^d));
%o A339458     until(ok,
%o A339458       p=smpr(b);
%o A339458       ok = 1;
%o A339458       listput(a,p,j);
%o A339458       if(p!=b,
%o A339458          d=log(log(p)/log(c))/log(j);
%o A339458          for(k=1,j-2,
%o A339458              b=floor(c^(k^d));
%o A339458              if(b!=a[k],
%o A339458                 ok=0;
%o A339458                 j=k;
%o A339458                 break;
%o A339458                );
%o A339458             );
%o A339458         );
%o A339458     );
%o A339458   );
%o A339458   default(realprecision, curprec);
%o A339458   return(contfrac(d));
%o A339458 } \\ _François Marques_, Dec 08 2020
%Y A339458 Cf. A339459, A339457, A338613, A338837, A338850.
%K A339458 nonn,cofr
%O A339458 1,4
%A A339458 _Bernard Montaron_, Dec 06 2020