A339458 -id:A339458 - cp's OEIS frontend

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

Bernard Montaron, Dec 06 2020

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)).
 n=1, a(1)=2, d=1
 n=n+1
 b=floor(2^(n^d))
 p=smpr(b)     (smallest prime >= b)
 If p=b, then a(n)=p, go to 1.
 d=log(log(p)/log(2))/log(n)
 a(n)=p
 k=1
 b=floor(2^(k^d))
 If b<>a(k) and b not prime, then p=smpr(b), n=k, go to 5.
 If b is prime, then a(k)=b
 If k  12.  go to 1.
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.

			1.5039285240695206335276890678975831991907388495811384290029993506576595475616...

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

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

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

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

Original entry on oeis.org

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

Offset: 1

Bernard Montaron, Dec 06 2020

nonn

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)).
  n=1, a(1)=2, d=1
  n=n+1
  b=floor(2^(n^d))
  p=smpr(b)     (smallest prime >= b)
  If p=b, then a(n)=p, go to 1.
  d=log(log(p)/log(2))/log(n)
  a(n)=p
  k=1
  b=floor(2^(k^d))
  If b<>a(k) and b not prime, then p=smpr(b), n=k, go to 5.
 If b is prime then a(k)=b
 If k12.  go to 1.
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.

			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.

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

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

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

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

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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.

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