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.

A091592 Numbers n such that there are no twin primes between n^2 and (n+1)^2.

Original entry on oeis.org

1, 9, 19, 26, 27, 30, 34, 39, 49, 53, 77, 122
Offset: 1

Views

Author

Hugo Pfoertner, Jan 25 2004

Keywords

Comments

Numbers n such that there is no pair of twin primes P, P+2 with n^2 < P < P+2 < n^2+2*n.
The first 7 terms of this sequence were given by Ernst Jung in a discussion in the Newsgroup de.sci.mathematik entitled "Primzahlen zwischen (2x-1)^2 und (2x+1)^2" (primes between ...and...) with other significant contributions from Hermann Kremer and Rainer Rosenthal. It is conjectured that there are no further terms beyond a(12)=122. This has been tested to 50000 by Robert G. Wilson v.
Tested up to 10^7 and found no such numbers. - Arkadiusz Wesolowski, Jul 11 2011

Examples

			9 is a term because no twin primes are found in the interval [9^2,10^2].

Links

Crossrefs

Cf. A091591, A036061, A036063, A113274.

Programs

  • Maple
    isA091592 := proc(n) local p; p := nextprime(n^2) ; q := nextprime(p) ; while q < n^2+2*n do if q-p = 2 then RETURN(false) ; fi; p :=q ; q := nextprime(p) ; od: RETURN(true) ; end: for n from 1 do if isA091592(n) then printf("%d ",n) ; fi; od: # R. J. Mathar, Aug 26 2008
  • Mathematica
    fQ[n_] := StringCount[ ToString@ PrimeQ[ Range[n^2, (n + 1)^2]], "True, False, True"] == 0; lst = {}; Do[ If[ fQ@n, AppendTo[lst, n]], {n, 25000}]

Extensions

Edited by N. J. A. Sloane, Aug 31 2008 at the suggestion of Pierre CAMI

A138205 Least number k such that n^2+k and n^2+k+2 are the smallest twin primes between squares n^2 and (n+1)^2, or 0 if there is no such k.

Original entry on oeis.org

0, 1, 2, 1, 4, 5, 10, 7, 0, 1, 16, 5, 10, 1, 2, 13, 22, 23, 0, 19, 20, 37, 40, 23, 16, 0, 0, 25, 16, 0, 58, 7, 2, 0, 4, 5, 58, 7, 0, 7, 16, 23, 22, 13, 2, 13, 28, 5, 0, 49, 56, 7, 0, 53, 94, 31, 2, 7, 46, 71, 46, 7, 32, 31, 4, 65, 28, 13, 26, 31, 58, 47, 88, 1, 14, 73, 0, 5, 28, 49, 8, 37
Offset: 1

Views

Author

T. D. Noe, Mar 06 2008

Keywords

Comments

Checking up to n=10^6, it appears that a(n)=0 for only n=1 and the n in A091592.

Examples

			a(7)=10 because (59,61) is the smallest twin-prime pair between 49 and 64.

Links

Crossrefs

Cf. A091591 (number of twin prime pairs between n^2 and (n+1)^2).

Programs

  • Mathematica
    Table[n2=n^2; k=1; While[k<2n+1 && !(PrimeQ[n2+k] && PrimeQ[n2+k+2]), k++ ]; If[k==2n+1,0,k], {n,10^4}]

A099154 Largest number k such that the interval [k^2,(k+1)^2] contains not more than n pairs of twin primes.

Original entry on oeis.org

122, 213, 502, 545, 922, 950, 749, 1098, 1330, 1450, 1634, 1623, 2135, 2110, 2177, 2244, 2760, 2413, 2556, 3280, 3454, 3211, 3740, 3540, 4104, 4096, 4391, 4457, 4592, 5309, 4758, 5720, 5747, 5295, 5902, 5456, 5920, 6395, 5810, 7007, 7109, 7450, 7540
Offset: 0

Views

Author

Hugo Pfoertner, Sep 30 2004

Keywords

Comments

The terms of this sequence are conjectural, even under the twin prime conjecture.

Examples

			a(1)=213 because the interval [213^2,214^2]=[45369,45796] contains one pair of twin primes (45587,45589) whereas all higher intervals are conjectured to contain at least two pairs of twin primes.
The interval [122^2,123^2]=[A091592(11)^2,(A091592(11)+1)^2] is conjectured to be the last interval between two consecutive squares containing no twin primes.

Links

Crossrefs

Cf. A091591 number of pairs of twin primes between n^2 and (n+1)^2, A091592 numbers n such that there are no twin primes between n^2 and (n+1)^2, A014574.
Showing 1-3 of 3 results.

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

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