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A091592 Numbers n such that there are no twin primes between n^2 and (n+1)^2. 6
1, 9, 19, 26, 27, 30, 34, 39, 49, 53, 77, 122 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

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(11)=122. This has been tested to 50000 by Robert G. Wilson v.

Tested up to 10^7 and found no such numbers. [From Arkadiusz Wesolowski, Jul 11 2011]

LINKS

Table of n, a(n) for n=1..12.

Hugo Pfoertner, Illustration of record gaps between pairs of twin primes.

Eric Weisstein's World of Mathematics, k-Tuple Conjecture.

Eric Weisstein's World of Mathematics, Twin Prime Conjecture.

EXAMPLE

a(1)=9 because no twin primes are found in the interval [9^2,10^2].

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: [From 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}]

CROSSREFS

Cf. A091591, A036061, A036063.

Sequence in context: A167529 A228610 A106677 * A174372 A145906 A090065

Adjacent sequences:  A091589 A091590 A091591 * A091593 A091594 A091595

KEYWORD

hard,nonn

AUTHOR

Hugo Pfoertner, Jan 25 2004

EXTENSIONS

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

STATUS

approved

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Last modified April 20 23:07 EDT 2014. Contains 240824 sequences.