A189025 Number of primes in the range (n - 2*sqrt(n), n].

0, 1, 2, 2, 3, 3, 4, 3, 2, 2, 3, 2, 3, 3, 2, 2, 3, 3, 4, 3, 3, 3, 3, 3, 3, 3, 3, 2, 3, 2, 3, 3, 3, 3, 2, 2, 3, 3, 3, 3, 4, 3, 4, 4, 3, 3, 4, 4, 4, 4, 4, 3, 4, 4, 4, 3, 3, 3, 3, 3, 4, 4, 3, 3, 3, 3, 4, 4, 4, 3, 4, 4, 5, 5, 5, 5, 4, 4, 4, 4, 4, 4, 5, 5, 5, 4, 4, 4, 5, 4, 4, 4, 3, 3, 3, 3, 4, 4, 3, 3, 4, 4, 5, 4, 4, 4, 5, 5, 6, 5, 5, 5, 6, 6, 6, 6, 6, 6, 5, 5, 5, 5, 5, 4, 4, 3, 4

Offset: 1

Author

T. D. Noe, Apr 15 2011

Keywords

• nonn
• easy

Comments

Note that the lower bound, n-2*sqrt(n), is excluded from the count and the upper range, n, is included. The only zero term appears to be a(1). See A189027 for special primes associated with this sequence. This sequence is related to Legendre's conjecture that there is a prime between consecutive squares.

Links

• T. D. Noe, Table of n, a(n) for n = 1..10000
• Wikipedia, Legendre's conjecture

Crossrefs

Cf. A188817, A189024, A189026.

Programs

Mathematica

cnt = 0; lastLower = -3; Table[lower = Floor[n - 2*Sqrt[n]]; If[lastLower < lower && PrimeQ[lower], cnt--]; lastLower = lower; If[PrimeQ[n], cnt++]; cnt, {n, 100}]
Table[PrimePi[n]-PrimePi[n-2Sqrt[n]],{n,130}] (* Harvey P. Dale, Feb 28 2023 *)

PARI

a(n)=if(nCharles R Greathouse IV, May 11 2011