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

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

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(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

Links

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

J. Korevaar, The prime-pair conjectures of Hardy and Littlewood, Indagationes Mathematicae, Volume 23, Issue 3, 2012, Pages 269-299.

A. Kourbatov, Maximal Gaps Between Prime k-Tuples: 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, k-Tuple 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 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}]

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