A339457 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.

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, 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, 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

Offset

1,2

Comments

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...

Algorithm to generate the smallest constant d and the associated prime number sequence a(n)=floor(2^(n^d)).

0. n=1, a(1)=2, d=1

1. n=n+1

2. b=floor(2^(n^d))

3. p=smpr(b) (smallest prime >= b)

4. If p=b, then a(n)=p, go to 1.

5. d=log(log(p)/log(2))/log(n)

6. a(n)=p

7. k=1

8. b=floor(2^(k^d))

9. If b<>a(k) and b not prime, then p=smpr(b), n=k, go to 5.

10. If b is prime, then a(k)=b

11. If k<n-1 then k=k+1, go to 8.

12. go to 1.

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.

Links

Table of n, a(n) for n=1..109.

Bernard Montaron, Exponential prime sequences, arXiv:2011.14653 [math.NT], 2020.

Example

1.5039285240695206335276890678975831991907388495811384290029993506576595475616...

Prog

(PARI) A339457(n=63, prec=200) = {
\\ returns the list of the first digits of the constant.
\\ the number of digits increases faster than n
  my(curprec=default(realprecision));
  default(realprecision, max(prec, curprec));
  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); );
  for(j=1, n-1,
    b=floor(c^(j^d));
    until(ok,
      p=smpr(b);
      ok = 1;
      listput(a, p, j);
      if(p!=b,
         d=log(log(p)/log(c))/log(j);
         for(k=1, j-2,
             b=floor(c^(k^d));
             if(b!=a[k],
                ok=0;
                j=k;
                break;
               );
            );
        );
    );
  );
  my(p=floor(-log(d-log(log(a[n-2])/log(c))/log(n-2))/log(10)) );
  default(realprecision, curprec);
  return(digits(floor(d*10^p), 10));
} \\ François Marques, Dec 08 2020

Crossrefs

Cf. A339459, A339458, A338613, A338837, A338850.

Sequence in context: A020856 A299621 A182567 * A300728 A321418 A225424

Adjacent sequences: A339454 A339455 A339456 * A339458 A339459 A339460

Keyword

nonn,cons

Author

Bernard Montaron, Dec 06 2020

Status

approved