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A338613 Numbers given by a(n) = 1 + floor(c^(n^1.5)) where c=2.2679962677... is the constant defined at A338837 6
2, 3, 11, 71, 701, 9467, 168599, 3860009, 111498091, 4002608003, 176359202639, 9437436701437, 607818993573569, 46744099128452807, 4262700354254812091, 458091929703695291747, 57691186909930154615407, 8471601990692484416847631, 1443868262009075144775972529 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,1

COMMENTS

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...

This exponential sequence of prime numbers grows very slowly compared to Mills' sequence for which each new term has 3 times more digits than the previous one. More than 60 terms (all prime numbers) can be easily calculated for the sequence described here which is quite remarkable for an exponential sequence.

Algorithm to compute the smallest constant 'c' and the associated prime number sequence a(n).

0.   n=0, a(0)=2, c=2, d=1.5

1.   n=n+1

2.   b=1+floor(c^(n^d))

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

4.   If p=b then a(n)=p, go to 1.

5.   c=(p-1)^(1/n^d)

6.   a(n)=p

7.   k=1

8.   b=1+floor(c^(k^d))

9.   If b<>a(k) then p=smpr(b), n=k, go to 5.

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

11. go to 1.

I propose the following generalization:  find the function f(n) with f(0)=0 and f(x)>x for x>=2 such that there exists a suitable positive constant c(f) giving the increasing prime sequence a(n)=1+floor(c^f(n)) with the smallest possible growth rate. Since a(0)=2, c(f)>=2.

LINKS

François Marques, Table of n, a(n) for n = 0..199

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

FORMULA

a(n) = 1 + floor(c^(n^1.5)) where c=2.2679962677...

PROG

(PARI)

c(n=40, prec=100)={

  my(curprec=default(realprecision));

  default(realprecision, max(prec, curprec));

  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); );

  for(j=1, n-1,

    b=1+floor(c^(j^d));

    until(ok,

      ok=1;

      p=smpr(b);

      listput(a, p, j+1);

      if(p!=b,

         c=(p-1)^(j^(-d));

         for(k=1, j-2,

             b=1+floor(c^(k^d));

             if(b!=a[k+1],

                ok=0;

                j=k;

                break;

               );

            );

        );

    );

  );

  default(realprecision, curprec);

  return(a);

} \\ François Marques, Nov 12 2020

CROSSREFS

Cf. A338837, A338850, A051021, A051254.

Sequence in context: A241811 A349518 A155187 * A109132 A279084 A008510

Adjacent sequences:  A338610 A338611 A338612 * A338614 A338615 A338616

KEYWORD

nonn

AUTHOR

Bernard Montaron, Nov 03 2020

STATUS

approved

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Last modified October 3 22:17 EDT 2022. Contains 357237 sequences. (Running on oeis4.)