

A091592


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


8



1, 9, 19, 26, 27, 30, 34, 39, 49, 53, 77, 122
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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 (2x1)^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.  Arkadiusz Wesolowski, Jul 11 2011


LINKS

Table of n, a(n) for n=1..12.
A. Kourbatov, Maximal Gaps Between Prime kTuples: A Statistical Approach, J. Int. Seq. 16 (2013) #13.5.2
Hugo Pfoertner, Illustration of record gaps between pairs of twin primes.
Eric Weisstein's World of Mathematics, kTuple Conjecture.
Eric Weisstein's World of Mathematics, Twin Prime Conjecture.


EXAMPLE

9 is a term 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 qp = 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}]


CROSSREFS

Cf. A091591, A036061, A036063, A113274.
Sequence in context: A228610 A106677 A350261 * A174372 A145906 A090065
Adjacent sequences: A091589 A091590 A091591 * A091593 A091594 A091595


KEYWORD

nonn,hard,more


AUTHOR

Hugo Pfoertner, Jan 25 2004


EXTENSIONS

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


STATUS

approved



