A195100 - OEIS

%I #36 Mar 23 2019 03:57:48

%S 1,11,21,25,28,33,66,122,140,142,188,307,322,349,1007,1052

%N Numbers n such that there are no primes between (n-1)*sqrt(n-1) and n*sqrt(n).

%C Cramér's conjecture implies that the sequence is finite. - _Robert Israel_, Aug 11 2014

%C No more terms up to 2*10^10. - _Jinyuan Wang_, Mar 22 2019

%H H. Cramér, <a href="http://matwbn.icm.edu.pl/ksiazki/aa/aa2/aa212.pdf">On the order of magnitude of the difference between consecutive prime numbers</a>, Acta Arith. 2 (1936), 23-46.

%F a(n+1) = A144140(n) + 1. - _Jinyuan Wang_, Mar 22 2019

%e a(1) = 1 because there are no numbers between (1-1)*sqrt(1-1) = 0 and 1*sqrt(1) = 1.

%e a(2) = 11 because (11-1)*sqrt(11-1) < (nonprimes 32,33,34,35,36) < 11*sqrt(11).

%p Primes:= select(isprime,{2,seq(2*i+1,i=1..10^6)}):

%p C:= map(p -> ceil(p^(2/3)), Primes);

%p {$1..max(C)} minus C; # _Robert Israel_, Aug 10 2014

%t Select[Range[5000], (PrimePi[# Sqrt[#]] - PrimePi[(# - 1)Sqrt[# - 1]]) == 0 &] (* _Alonso del Arte_, Sep 09 2011 *)

%t Join[{1},Flatten[Position[Partition[Table[PrimePi[n Sqrt[n]],{n,1100}], 2,1], _?(#[[2]]-#[[1]]==0&),1,Heads->False]]+1] (* _Harvey P. Dale_, May 11 2018 *)

%o (PARI) for(n=1,2*10^6,if(#primes([(n-1)*sqrt(n-1),n*sqrt(n)])==0,print1(n,", "))) \\ _Derek Orr_, Aug 10 2014

%o (PARI) isok(n) = {k=floor((n-1)*sqrt(n-1))+1;while(!isprime(k),k++);k>n*sqrt(n);} \\ _Jinyuan Wang_, Mar 22 2019

%Y Cf. A001223, A144140, A194852, A195008.

%K nonn,more

%O 1,2

%A _Juri-Stepan Gerasimov_, Sep 09 2011