A189024 Number of primes in the range (n - sqrt(n), n].

0, 1, 2, 1, 2, 1, 2, 1, 1, 1, 1, 1, 2, 2, 1, 1, 2, 1, 2, 2, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 1, 1, 2, 1, 1, 1, 2, 2, 3, 2, 2, 2, 3, 2, 2, 2, 1, 1, 2, 2, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 1, 2, 2, 3, 3, 3, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 2, 2, 1, 1, 2, 2, 3, 3, 3, 3, 4, 3, 4, 4, 4, 3, 4, 3, 3, 3, 3, 2, 2, 1, 1, 1, 1, 1, 0, 0, 1

Offset: 1

Author

T. D. Noe, Apr 15 2011

Keywords

• nonn

Comments

Note that the lower bound, n-sqrt(n), is excluded from the count and the upper range, n, is included. The last zero term appears to be a(126). See A189026 for special primes associated with this sequence. This sequence is related to Oppermann's conjecture that for any k > 1 there is a prime between k^2 - k and k^2.

Links

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

Crossrefs

Cf. A188817, A189025, A189027.

Programs

Mathematica

cnt = 0; lastLower = 0; Table[lower = Floor[n - Sqrt[n]]; If[lastLower < lower && PrimeQ[lower], cnt--]; lastLower = lower; If[PrimeQ[n], cnt++]; cnt, {n, 100}]
Table[PrimePi[n]-PrimePi[n-Sqrt[n]],{n,130}] (* Harvey P. Dale, Mar 26 2023 *)