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A124588 Primes p such that q - p <= 2, where q is the next prime after p.

%I A124588 #35 May 13 2022 09:52:33
%S A124588 2,3,5,11,17,29,41,59,71,101,107,137,149,179,191,197,227,239,269,281,
%T A124588 311,347,419,431,461,521,569,599,617,641,659,809,821,827,857,881,1019,
%U A124588 1031,1049,1061,1091,1151,1229,1277,1289,1301,1319,1427,1451,1481,1487,1607
%N A124588 Primes p such that q - p <= 2, where q is the next prime after p.
%C A124588 Consists of 2 together with the lower members of twin primes, A001359. See the latter entry for references.
%C A124588 "Assuming certain (admittedly difficult) conjectures on the distribution of primes in arithmetic progressions, [Goldston-Pintz-Yildirim] prove the existence of infinitely many prime pairs that differ at most by 16." - Soundararajan
%C A124588 Lesser of twin primes together with 2; union with A029710 gives A124589. - _Reinhard Zumkeller_, Dec 23 2006
%C A124588 Primes p such that either p + 3/2 +- 1/2 is prime. - _Juri-Stepan Gerasimov_, Jan 29 2010
%C A124588 The prime differences of 2 primes (without repetition). - _Juri-Stepan Gerasimov_, Jun 01 2010, Jun 08 2010
%C A124588 Numbers k such that sigma(k*(k+2)) = (k+1)*(k+3). - _Wesley Ivan Hurt_, May 08 2022
%H A124588 K. Soundararajan, <a href="https://doi.org/10.1090/S0273-0979-06-01142-6">Small gaps between prime numbers: the work of Goldston-Pintz-Yildirim</a>, Bull. Amer. Math. Soc., 44 (2007), 1-18.
%H A124588 <a href="/index/Pri#gaps">Index entries for primes, gaps between</a>
%t A124588 Transpose[Select[Partition[Prime[Range[300]],2,1],#[[2]]-#[[1]]<3&]] [[1]] (* _Harvey P. Dale_, Feb 11 2015 *)
%o A124588 (PARI) twinl(n) = { c=0; x=1; while(c<n, if(isprime(prime(x)+2), c++); x++; ); return(prime(x-1)) }
%o A124588 print1(2", "); (for(x=1, 200, print1(twinl(x)", "))) \\ _Cino Hilliard_, Mar 29 2008
%Y A124588 Cf. A001359.
%K A124588 nonn
%O A124588 1,1
%A A124588 _N. J. A. Sloane_, Dec 19 2006; edited May 15 2008 at the suggestion of _R. J. Mathar_