%I #32 Oct 23 2023 10:45:03
%S 1,9,19,26,27,30,34,39,49,53,77,122
%N Numbers n such that there are no twin primes between n^2 and (n+1)^2.
%C Numbers n such that there is no pair of twin primes P, P+2 with n^2 < P < P+2 < n^2+2*n.
%C 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_.
%C Tested up to 10^7 and found no such numbers. - _Arkadiusz Wesolowski_, Jul 11 2011
%H J. Korevaar, <a href="https://doi.org/10.1016/j.indag.2011.12.001">The prime-pair conjectures of Hardy and Littlewood</a>, Indagationes Mathematicae, Volume 23, Issue 3, 2012, Pages 269-299.
%H A. Kourbatov, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL16/Kourbatov/kourbatov3.html">Maximal Gaps Between Prime k-Tuples: A Statistical Approach</a>, J. Int. Seq. 16 (2013) #13.5.2
%H Hugo Pfoertner, <a href="http://www.randomwalk.de/sequences/a091592.pdf">Illustration of record gaps between pairs of twin primes.</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/k-TupleConjecture.html">k-Tuple Conjecture</a>.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/TwinPrimeConjecture.html">Twin Prime Conjecture</a>.
%e 9 is a term because no twin primes are found in the interval [9^2,10^2].
%p 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
%t 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}]
%Y Cf. A091591, A036061, A036063, A113274.
%K nonn,hard,more
%O 1,2
%A _Hugo Pfoertner_, Jan 25 2004
%E Edited by _N. J. A. Sloane_, Aug 31 2008 at the suggestion of _Pierre CAMI_