cp's OEIS Frontend

A338837 Decimal expansion of the smallest positive number 'c' such that numbers of the form 1+floor(c^(n^1.5)) for all n >= 0 are distinct primes.

%I A338837 #21 Nov 17 2020 14:10:37
%S A338837 2,2,6,7,9,9,6,2,6,7,7,0,6,7,2,4,2,4,7,3,2,8,5,5,3,2,8,0,7,2,5,3,7,1,
%T A338837 7,7,4,5,2,7,0,4,2,2,5,4,4,0,0,8,1,8,7,7,2,2,7,5,5,9,0,8,2,9,0,5,0,7,
%U A338837 8,3,7,4,0,7,5,1,4,6,9,5,7,3,5,7,2,1,7,3,8,3,6,2,9,0,9,9,2,5,7,0,0,4,2,7,3,1,5,8,7,3,1,7,1,1,5,7,6,5,8,8,1,9,3,4,0,9,7,2,8,1,1,3,9,0
%N A338837 Decimal expansion of the smallest positive number 'c' such that numbers of the form 1+floor(c^(n^1.5)) for all n >= 0 are distinct primes.
%C A338837 Assuming Cramer's conjecture on largest prime gaps, it can be proved that there exists at least one constant 'c' such that all a(n) are primes for n as large as required. The constant giving the smallest growth rate is c=2.2679962677067242473285532807253717745270422544...
%C A338837 Algorithm to generate the smallest constant 'c' and the associated prime number sequence a(n)=1+floor(c^(n^1.5)).
%C A338837 0.   n=0, a(0)=2, c=2, d=1.5
%C A338837 1.   n=n+1
%C A338837 2.   b=1+floor(c^(n^d))
%C A338837 3.   p=smpr(b)     smallest prime >= b
%C A338837 4.   If p=b then a(n)=p, go to 1.
%C A338837 5.   c=(p-1)^(1/n^d)
%C A338837 6.   a(n)=p
%C A338837 7.   k=1
%C A338837 8.   b=1+floor(c^(k^d))
%C A338837 9.   If b<>a(k) then p=smpr(b), n=k, go to 5.
%C A338837 10. If k<n-1 then k=k+1, go to 8.
%C A338837 11. go to 1.
%C A338837 The precision of 'c' with the 135 digits listed above is sufficient to calculate the first 50 terms of the prime sequence. The prime number given by the term of index n=49 has 121 decimal digits.
%e A338837 2.26799626770672424732855328072537177452704225440081877227559082905078374075...
%o A338837 (PARI)
%o A338837 c(n=40, prec=100)={
%o A338837   my(curprec=default(realprecision));
%o A338837   default(realprecision, max(prec, curprec));
%o A338837   my(a=List([2]), d=1.5, c=2.0, b, p, ok, smpr(b)=my(p=b); while(!isprime(p), p=nextprime(p+1)); return(p); );
%o A338837   for(j=1, n-1,
%o A338837     b=1+floor(c^(j^d));
%o A338837     until(ok,
%o A338837       ok=1;
%o A338837       p=smpr(b);
%o A338837       listput(a,p,j+1);
%o A338837       if(p!=b,
%o A338837          c=(p-1)^(j^(-d));
%o A338837          for(k=1,j-2,
%o A338837              b=1+floor(c^(k^d));
%o A338837              if(b!=a[k+1],
%o A338837                 ok=0;
%o A338837                 j=k;
%o A338837                 break;
%o A338837                );
%o A338837             );
%o A338837         );
%o A338837     );
%o A338837   );
%o A338837   default(realprecision, curprec);
%o A338837   return(c);
%o A338837 };
%o A338837 digits(floor(c(55,200)*10^50))
%o A338837 \\ _François Marques_, Nov 17 2020
%Y A338837 Cf. A338613, A338850.
%K A338837 nonn,cons
%O A338837 1,1
%A A338837 _Bernard Montaron_, Nov 11 2020