A195100 Numbers n such that there are no primes between (n-1)*sqrt(n-1) and n*sqrt(n).

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

Offset

1,2

Comments

Cramér's conjecture implies that the sequence is finite. - Robert Israel, Aug 11 2014

No more terms up to 2*10^10. - Jinyuan Wang, Mar 22 2019

Links

Table of n, a(n) for n=1..16.

H. Cramér, On the order of magnitude of the difference between consecutive prime numbers, Acta Arith. 2 (1936), 23-46.

Formula

a(n+1) = A144140(n) + 1. - Jinyuan Wang, Mar 22 2019

Example

a(1) = 1 because there are no numbers between (1-1)*sqrt(1-1) = 0 and 1*sqrt(1) = 1.
a(2) = 11 because (11-1)*sqrt(11-1) < (nonprimes 32,33,34,35,36) < 11*sqrt(11).

Maple

Primes:= select(isprime, {2, seq(2*i+1, i=1..10^6)}):
C:= map(p -> ceil(p^(2/3)), Primes);
{$1..max(C)} minus C; # Robert Israel, Aug 10 2014

Mathematica

Select[Range[5000], (PrimePi[# Sqrt[#]] - PrimePi[(# - 1)Sqrt[# - 1]]) == 0 &] (* Alonso del Arte, Sep 09 2011 *)
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 *)

Prog

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

Crossrefs

Cf. A001223, A144140, A194852, A195008.

Sequence in context: A350766 A189226 A261409 * A125164 A101223 A109686

Adjacent sequences: A195097 A195098 A195099 * A195101 A195102 A195103

Keyword

nonn,more

Author

Juri-Stepan Gerasimov, Sep 09 2011

Status

approved