%I #27 Jan 01 2022 09:53:38
%S 1,5,0,3,9,2,8,5,2,4,0,6,9,5,2,0,6,3,3,5,2,7,6,8,9,0,6,7,8,9,7,5,8,3,
%T 1,9,9,1,9,0,7,3,8,8,4,9,5,8,1,1,3,8,4,2,9,0,0,2,9,9,9,3,5,0,6,5,7,6,
%U 5,9,5,4,7,5,6,1,6,3,0,5,7,6,4,3,1,7,1,0,1,8,9,0,8,0,8,8,6,5,2,2,4,6,8,7,4,0,1,3,0
%N Decimal expansion of the smallest positive number d such that numbers of the sequence floor(2^(n^d)) are distinct primes for all n>=1.
%C Assuming Cramer's conjecture on prime gaps, it can be proved that there exists at least one constant d such that all floor(2^(n^d)) are primes for n>=1 as large as required. The constant giving the smallest growth rate is d=1.503928524069520633527689067897583199190738...
%C Algorithm to generate the smallest constant d and the associated prime number sequence a(n)=floor(2^(n^d)).
%C 0. n=1, a(1)=2, d=1
%C 1. n=n+1
%C 2. b=floor(2^(n^d))
%C 3. p=smpr(b) (smallest prime >= b)
%C 4. If p=b, then a(n)=p, go to 1.
%C 5. d=log(log(p)/log(2))/log(n)
%C 6. a(n)=p
%C 7. k=1
%C 8. b=floor(2^(k^d))
%C 9. If b<>a(k) and b not prime, then p=smpr(b), n=k, go to 5.
%C 10. If b is prime, then a(k)=b
%C 11. If k<n-1 then k=k+1, go to 8.
%C 12. go to 1.
%C 112 decimal digits of d are sufficient to calculate the first 50 terms of the prime sequence. The prime number given by the term of index n=50 has 109 decimal digits.
%H Bernard Montaron, <a href="https://arxiv.org/abs/2011.14653">Exponential prime sequences</a>, arXiv:2011.14653 [math.NT], 2020.
%e 1.5039285240695206335276890678975831991907388495811384290029993506576595475616...
%o (PARI) A339457(n=63, prec=200) = {
%o \\ returns the list of the first digits of the constant.
%o \\ the number of digits increases faster than n
%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 my(p=floor(-log(d-log(log(a[n-2])/log(c))/log(n-2))/log(10)) );
%o default(realprecision, curprec);
%o return(digits(floor(d*10^p),10));
%o } \\ _François Marques_, Dec 08 2020
%Y Cf. A339459, A339458, A338613, A338837, A338850.
%K nonn,cons
%O 1,2
%A _Bernard Montaron_, Dec 06 2020