cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

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

Original entry on oeis.org

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

Views

Author

T. D. Noe, Apr 15 2011

Keywords

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

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

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

Original entry on oeis.org

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

Views

Author

T. D. Noe, Apr 15 2011

Keywords

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

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 *)

A189026 There appear to be at least n primes in the range (x-sqrt(x), x] for all x >= a(n).

Original entry on oeis.org

127, 1367, 2531, 2539, 6007, 7457, 10061, 10847, 23531, 35797, 35801, 38557, 44497, 47111, 69767, 69809, 88321, 107687, 110419, 110431, 113723, 127217, 250673, 250681, 250687, 250703, 268487, 268493, 286381, 286393, 302563, 302567, 360947, 369821, 405199
Offset: 1

Views

Author

T. D. Noe, Apr 15 2011

Keywords

Comments

These terms exist only if a strong form of Oppermann's conjecture that for any k>1 there is a prime between k^2-k and k^2 is true. Note that every term is prime. Sequence A189024 gives the number of primes in the range (x-sqrt(x), x]. The index of the prime a(n), that is, primepi(a(n)), is approximately (2.4*n)^2. These primes are generated in a manner similar to the Ramanujan primes (A104272).

Links

Crossrefs

Cf. A189025, A189027.
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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