A339459 Prime numbers a(n) = floor(2^(n^d)) for all n>=1 where d=1.5039285240... is the constant defined at A339457.

2, 7, 37, 263, 2437, 28541, 414893, 7368913, 157859813, 4035572951, 122006926709, 4328504865941, 178988464493359, 8575347401843113, 473485756611713633, 29985730185033339911, 2168685169398896331137, 178419507110725228550743

Offset

1,1

Comments

Assuming Cramer's conjecture on prime gaps is true, it can be proved that there exists at least one constant d such that all terms of the sequence are primes. 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 decimals 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..18.

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

Formula

a(n) = floor(2^(n^d)) where d=1.5039285240...

Example

This example illustrates the importance of doing full precision calculations: a(19) = floor(2^(19^d)) = floor(2^83.7826351429215150692195114432) = 16637432012996855576590853. Here, the precision required on the exponent of 2 is 28 decimals in order to obtain the correct value for a(19). And the precision required keeps increasing with the index value n.

Prog

(PARI) A339459(n=30, prec=100) = {
\\ if precision is large enough, returns the list of first n terms of the sequence
  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;
               );
            );
        );
    );
  );
  default(realprecision, curprec);
  return(a);
} \\ François Marques, Dec 08 2020

Crossrefs

Cf. A339457, A339458, A338613, A338837, A338850.

Sequence in context: A321087 A072597 A322140 * A125515 A135920 A001515

Adjacent sequences: A339456 A339457 A339458 * A339460 A339461 A339462

Keyword

nonn

Author

Bernard Montaron, Dec 06 2020

Status

approved