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A075321 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), (53, 67) ... This is the sequence of the first member of every pair. 3

%I

%S 3,7,13,23,37,17,53,43,61,83,109,73,101,139,41,149,157,137,113,193,

%T 197,179,211,229,263,199,227,107,331,293,311,283,241,269,349,359,383,

%U 367,401,317,379,439,491,421,409,449,463,467

%N 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), (53, 67) ... This is the sequence of the first member of every pair.

%C Question: Is every prime a member of some pair?

%C If the distance between the prime pairs is not required to be 2n, we get A031215. - _R. J. Mathar_, Nov 26 2014

%C a(n) = A075323(2*n-1).

%H Reinhard Zumkeller, <a href="/A075321/b075321.txt">Table of n, a(n) for n = 1..10000</a>

%e a(4)=23: For the 4th pair though 17 is the smallest prime not occurring earlier, 17+8 = 25 is not a prime and 23 + 8 = 31 is a prime.

%p A075321p := proc(n)

%p option remember;

%p local prevlist,i,p,q ;

%p if n = 1 then

%p return [3,5];

%p else

%p prevlist := [seq(op(procname(i)),i=1..n-1)] ;

%p for i from 2 do

%p p := ithprime(i) ;

%p if not p in prevlist then

%p q := p+2*n ;

%p if isprime(q) and not q in prevlist then

%p return [p,q] ;

%p end if;

%p end if;

%p end do:

%p end if;

%p end proc:

%p A075321 := proc(n)

%p op(1,A075321p(n)) ;

%p end proc:

%p seq(A075321(n),n=1..60) ; # _R. J. Mathar_, Nov 26 2014

%t 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 A075321 [n_] := A075321p[n][[1]];

%t Array[A075321, 50] (* _Jean-Fran├žois Alcover_, Feb 12 2018, translated from _R. J. Mathar_'s program *)

%o (Haskell)

%o a075321 = a075323 . subtract 1 . (* 2)

%o -- _Reinhard Zumkeller_, Nov 29 2014

%Y Cf. A075322, A075323.

%K nonn

%O 1,1

%A _Amarnath Murthy_, Sep 14 2002

%E Corrected by _R. J. Mathar_, Nov 26 2014

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Last modified August 2 12:30 EDT 2021. Contains 346424 sequences. (Running on oeis4.)