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A075323 Pair the odd primes so that the n-th pair is (p, p+2n) where p is the smallest prime not included earlier such that p and p+2n are primes and p+2n also does not occur earlier: (3, 5), (7, 11), (13, 19), (23, 31), (37, 47), (17, 29), ... This lists the successive pairs in order.

%I A075323 #21 Feb 12 2018 11:03:12
%S A075323 3,5,7,11,13,19,23,31,37,47,17,29,53,67,43,59,61,79,83,103,109,131,73,
%T A075323 97,101,127,139,167,41,71,149,181,157,191,137,173,113,151,193,233,197,
%U A075323 239,179,223,211,257,229,277,263,313,199
%N A075323 Pair the odd primes so that the n-th pair is (p, p+2n) where p is the smallest prime not included earlier such that p and p+2n are primes and p+2n also does not occur earlier: (3, 5), (7, 11), (13, 19), (23, 31), (37, 47), (17, 29), ... This lists the successive pairs in order.
%C A075323 Question: Is every odd prime a member of some pair?
%C A075323 2683 = A065091(388) seems to be missing, as presumably A247233(388)=0; but if A247233(n) > 0: a(A247233(n)) = A065091(n) = A000040(n+1). - _Reinhard Zumkeller_, Nov 29 2014
%H A075323 Reinhard Zumkeller, <a href="/A075323/b075323.txt">Table of n, a(n) for n = 1..10000</a>
%p A075323 # A075321p implemented in A075321
%p A075323 A075323 := proc(n)
%p A075323     if type(n,'odd') then
%p A075323         op(1,A075321p((n+1)/2)) ;
%p A075323     else
%p A075323         op(2,A075321p(n/2)) ;
%p A075323     end if;
%p A075323 end proc:
%p A075323 seq(A075323(n),n=1..60) ; # _R. J. Mathar_, Nov 26 2014
%t A075323 A075321p[n_] := A075321p[n] = Module[{prevlist, i, p, q}, If[n == 1, Return[{3, 5}], prevlist = Array[A075321p, n - 1] // Flatten]; For[i = 2, True, i++, p = Prime[i]; If[FreeQ[prevlist, p], q = p + 2*n; If[PrimeQ[q] && FreeQ[prevlist, q], Return[{p, q}]]]]];
%t A075323 A075323[n_] := If[OddQ[n], A075321p[(n+1)/2][[1]], A075321p[n/2][[2]]];
%t A075323 Array[A075323, 50] (* _Jean-François Alcover_, Feb 12 2018, after _R. J. Mathar_ *)
%o A075323 (Haskell)
%o A075323 import Data.List ((\\))
%o A075323 a075323 n = a075323_list !! (n-1)
%o A075323 a075323_list = f 1 []  $ tail a000040_list where
%o A075323    f k ys qs = g qs where
%o A075323      g (p:ps) | a010051' pk == 0 || pk `elem` ys = g ps
%o A075323               | otherwise = p : pk : f (k + 1) (p:pk:ys) (qs \\ [p, pk])
%o A075323               where pk = p + 2 * k
%o A075323 -- _Reinhard Zumkeller_, Nov 29 2014
%Y A075323 Cf. A075321, A075322.
%Y A075323 Cf. A010051, A000040, A065091, A247233.
%K A075323 nonn
%O A075323 1,1
%A A075323 _Amarnath Murthy_, Sep 14 2002
%E A075323 Corrected by _R. J. Mathar_, Nov 26 2014