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A020483 Least prime p such that p+2n is also prime.

%I A020483 #66 Sep 23 2020 04:20:39
%S A020483 2,3,3,5,3,3,5,3,3,5,3,7,5,3,3,7,5,3,5,3,3,5,3,7,5,3,7,5,3,3,7,5,3,5,
%T A020483 3,3,7,5,3,5,3,7,5,3,13,7,5,3,5,3,3,5,3,3,5,3,19,13,11,13,7,5,3,5,3,7,
%U A020483 5,3,3,11,11,7,5,3,3,7,5,3,7,5,3,5,3,7,5,3,7,5,3,3,11,11,7,5,3,3,5,3,3,13,11,31,7
%N A020483 Least prime p such that p+2n is also prime.
%C A020483 It is conjectured that a(n) always exists. a(n) has been computed for n < 5 * 10^11, with largest value a(248281210271) = 3307. - _Jens Kruse Andersen_, Nov 28 2004
%C A020483 If a(n) = a(n+1) = k, then 2*n + k and 2*(n+1) + k are twin primes. - _Ya-Ping Lu_, Sep 22 2020
%H A020483 T. D. Noe, <a href="/A020483/b020483.txt">Table of n, a(n) for n = 0..10000</a>
%H A020483 Jens Kruse Andersen, <a href="http://groups.yahoo.com/group/primenumbers/message/15641">Prime gaps (not necessarily consecutive)</a>, Yahoo! group "primenumbers", Nov 26 2004.
%H A020483 Jens Kruse Andersen, Mike Oakes, Ed Pegg Jr, <a href="/A020483/a020483.txt">Prime gaps (not necessarily consecutive)</a>, digest of 5 messages in primenumbers Yahoo group, Nov 26 - Nov 27, 2004.
%F A020483 If a(n) exists, a(n) < 2n, which of course is a great overestimate. - _T. D. Noe_, Jul 16 2002
%F A020483 a(n) = A087711(n) - n. - _Zak Seidov_, Nov 28 2007
%F A020483 a(n) = A020484(n) - 2n. - _Zak Seidov_, May 29 2014
%F A020483 a(n) = 2 if and only if n = 0. - _Alonso del Arte_, Mar 14 2018
%e A020483 Given n = 2, we see that 2 + 2n = 6 = 2 * 3, but 3 + 2n = 7, which is prime, so a(2) = 3.
%e A020483 Given n = 3, we see that 2 + 2n = 8 = 2^3 and 3 + 2n = 9 = 3^2, but 5 + 2n = 11, which is prime, so a(3) = 5.
%p A020483 A020483 := proc(n)
%p A020483     local p;
%p A020483     p := 2;
%p A020483     while true do
%p A020483         if isprime(p+2*n) then
%p A020483             return p;
%p A020483         end if;
%p A020483         p := nextprime(p) ;
%p A020483     end do:
%p A020483 end proc:
%p A020483 seq(A020483(n),n=0..40); # _R. J. Mathar_, Sep 23 2016
%t A020483 Table[j = 1; found = False; While[!found, j++; found = PrimeQ[Prime[j] + 2i]]; Prime[j], {i, 200}]
%t A020483 leastPrimep2n[n_] := Block[{k = 1, p, q = 2 n}, While[p = Prime@k; !PrimeQ[p + q], k++]; p]; Array[leastPrimep2n, 102] (* _Robert G. Wilson v_, Mar 26 2008 *)
%o A020483 (PARI) a(n)=forprime(p=2,,if(isprime(p+2*n), return(p))) \\ _Charles R Greathouse IV_, Mar 19 2014
%o A020483 (Haskell)
%o A020483 a020483 n = head [p | p <- a000040_list, a010051' (p + 2 * n) == 1]
%o A020483 -- _Reinhard Zumkeller_, Nov 29 2014
%o A020483 (GAP) P:=Filtered([1..10000],IsPrime);;
%o A020483 a:=List(List([0..110],n->Filtered(P,i->IsPrime(i+2*n))),Minimum); # _Muniru A Asiru_, Mar 26 2018
%Y A020483 Cf. A087711, A101042, A101043, A101044, A101045, A101046.
%Y A020483 Cf. A101045, A239392 (record values).
%Y A020483 Cf. A000040, A010051, A020484, A237055.
%Y A020483 It is likely that A054906 is an identical sequence, although this seems to have not yet been proved. - _N. J. A. Sloane_, Feb 06 2017
%K A020483 nonn
%O A020483 0,1
%A A020483 _David W. Wilson_
%E A020483 a(0)=2 added by _N. J. A. Sloane_, Apr 25 2015