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 1-6 of 6 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 *)

A192226 Numbers n such that all integers in the interval (n-2*sqrt(sqrt(n)), n] are composite.

Original entry on oeis.org

1, 28, 36, 96, 120, 121, 122, 123, 124, 125, 126, 146, 147, 148, 189, 190, 207, 208, 209, 210, 219, 220, 221, 222, 249, 250, 292, 302, 303, 304, 305, 306, 326, 327, 328, 329, 330, 346, 477, 478, 519, 520, 533, 534, 535, 536, 537, 538, 539, 540, 630, 672
Offset: 1

Views

Author

Juri-Stepan Gerasimov, Jun 26 2011

Keywords

Comments

a(14432) = 191913030 is probably the last term. Any further terms must be greater than 1.5 * 10^18. - Charles R Greathouse IV, Jun 30 2011

Links

Crossrefs

Cf. A188817, A189025, A008995.

Programs

  • Mathematica
    Select[Range[700],NextPrime[#-2Sqrt[Sqrt[#]]]>#&] (* Harvey P. Dale, Jul 27 2011 *)
  • PARI
    is(n)=for(k=0,sqrtnint(16*n-1,4),if(isprime(n-k), return(0))); 1 \\ Charles R Greathouse IV, Aug 26 2015

Extensions

a(12)-a(13) added, a(53)-a(56) corrected by Charles R Greathouse IV, Jun 30 2011

A192231 Numbers k without prime numbers in the range (k-3*sqrt(sqrt(k)), k].

Original entry on oeis.org

1, 123, 124, 125, 126, 306, 330, 538, 539, 540, 904, 905, 906, 1147, 1148, 1149, 1150, 1346, 1347, 1348, 1349, 1350, 1351, 1352, 1353, 1354, 1355, 1356, 1357, 1358, 1359, 1360, 1689, 1690, 1691, 1692, 1971, 1972, 2200, 2201
Offset: 1

Views

Author

Juri-Stepan Gerasimov, Jun 26 2011

Keywords

Comments

a(934) = 20831532 is probably the last term. Any further terms must be greater than 1.5 * 10^18. - Charles R Greathouse IV, Jun 27 2011

Examples

			a(2)=123 because (123-3*sqrt(sqrt(123)), 123]=(123-9,9907.., 123]=(113,0092.., 123].

Links

Crossrefs

Subsequence of A192226.
Cf. A001223, A188817, A189025, A192926.

Programs

Extensions

a(2) added by Alonso del Arte, Jun 27 2011

A192227 Number of primes in the range (n - 2*sqrt(sqrt(n)), n].

Original entry on oeis.org

0, 1, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 2, 2, 1, 1, 2, 1, 2, 2, 2, 1, 2, 1, 1, 1, 1, 0, 1, 1, 2, 2, 2, 1, 1, 0, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 2, 2, 2, 1
Offset: 1

Views

Author

Juri-Stepan Gerasimov, Jun 26 2011

Keywords

Comments

a(n) is probably positive for all n > 191913030. - Charles R Greathouse IV, Jul 01 2011

Crossrefs

Cf. A188817, A189025, A192226.

Programs

  • Maple
    A192227 := proc(n) local nhi, nlo ; nhi := n ; nlo := floor( n-2*root[4](n)) ; numtheory[pi](nhi)-numtheory[pi](nlo) ; end proc; # R. J. Mathar, Jul 12 2011
  • Mathematica
    Table[PrimePi[n]-PrimePi[n-2*Sqrt[Sqrt[n]]],{n,90}] (* Harvey P. Dale, Feb 24 2018 *)

Formula

Conjecturally, a(n) ~ 2*n^(1/4)/log n. - Charles R Greathouse IV, Jul 01 2011

A192360 Numbers k such that number of primes in the range (k-sqrt(k), k) is equal to number of primes in the range (k, k+sqrt(k)).

Original entry on oeis.org

1, 4, 5, 6, 9, 12, 15, 17, 18, 19, 22, 25, 30, 35, 42, 51, 53, 54, 59, 60, 61, 64, 67, 68, 69, 72, 76, 77, 78, 81, 82, 83, 88, 89, 92, 104, 105, 106, 120, 132, 133, 134, 135, 136, 143, 144, 149, 150, 151, 152, 153, 154, 157, 161, 163, 164, 165, 166
Offset: 1

Views

Author

Juri-Stepan Gerasimov, Jun 28 2011

Keywords

Crossrefs

Cf. A188817, A192221.

Programs

  • Maple
    isA192360 := proc(n) plow := floor(n-sqrt(n)) ; phi := ceil(n+sqrt(n)) ; plow := numtheory[pi](n-1)-numtheory[pi](plow) ; phi := numtheory[pi] (phi-1)-numtheory[pi](n) ; plow = phi ; end proc:
for n from 1 to 200 do if isA192360(n) then printf("%d,",n) ; end if; end do: # R. J. Mathar, Jul 02 2011
  • PARI
    isA192360(n)=my(s=sqrtint(n));2*primepi(n)-isprime(n)==if(n==s^2,primepi(n-s)+primepi(n+s-1),primepi(n-s-1)+primepi(n+s))

Extensions

3 removed by Charles R Greathouse IV, Jun 29 2011
Showing 1-6 of 6 results.

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