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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.

%I A338613 #59 Jun 02 2025 15:23:34
%S A338613 2,3,11,71,701,9467,168599,3860009,111498091,4002608003,176359202639,
%T A338613 9437436701437,607818993573569,46744099128452807,4262700354254812091,
%U A338613 458091929703695291747,57691186909930154615407,8471601990692484416847631,1443868262009075144775972529
%N A338613 Numbers given by a(n) = 1 + floor(c^(n^1.5)) where c=2.2679962677... is the constant defined at A338837.
%C A338613 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...
%C A338613 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.
%C A338613 Algorithm to compute the smallest constant 'c' and the associated prime number sequence a(n).
%C A338613 0.   n=0, a(0)=2, c=2, d=1.5
%C A338613 1.   n=n+1
%C A338613 2.   b=1+floor(c^(n^d))
%C A338613 3.   p=smpr(b)     smallest prime >= b
%C A338613 4.   If p=b then a(n)=p, go to 1.
%C A338613 5.   c=(p-1)^(1/n^d)
%C A338613 6.   a(n)=p
%C A338613 7.   k=1
%C A338613 8.   b=1+floor(c^(k^d))
%C A338613 9.   If b<>a(k) then p=smpr(b), n=k, go to 5.
%C A338613 10. If k<n-1 then k=k+1, go to 8.
%C A338613 11. go to 1.
%C A338613 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.
%H A338613 François Marques, <a href="/A338613/b338613.txt">Table of n, a(n) for n = 0..199</a>
%H A338613 Bernard Montaron, <a href="https://arxiv.org/abs/2011.14653">Exponential prime sequences</a>, arXiv:2011.14653 [math.NT], 2020.
%F A338613 a(n) = 1 + floor(c^(n^1.5)) where c=2.2679962677...
%o A338613 (PARI)
%o A338613 c(n=40, prec=100)={
%o A338613   my(curprec=default(realprecision));
%o A338613   default(realprecision, max(prec, curprec));
%o A338613   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); );
%o A338613   for(j=1, n-1,
%o A338613     b=1+floor(c^(j^d));
%o A338613     until(ok,
%o A338613       ok=1;
%o A338613       p=smpr(b);
%o A338613       listput(a,p,j+1);
%o A338613       if(p!=b,
%o A338613          c=(p-1)^(j^(-d));
%o A338613          for(k=1,j-2,
%o A338613              b=1+floor(c^(k^d));
%o A338613              if(b!=a[k+1],
%o A338613                 ok=0;
%o A338613                 j=k;
%o A338613                 break;
%o A338613                );
%o A338613             );
%o A338613         );
%o A338613     );
%o A338613   );
%o A338613   default(realprecision, curprec);
%o A338613   return(a);
%o A338613 } \\ _François Marques_, Nov 12 2020
%Y A338613 Cf. A338837, A338850, A051021, A051254.
%K A338613 nonn
%O A338613 0,1
%A A338613 _Bernard Montaron_, Nov 03 2020