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, 7, 13, 23, 37, 17, 53, 43, 61, 83, 109, 73, 101, 139, 41, 149, 157, 137, 113, 193, 197, 179, 211, 229, 263, 199, 227, 107, 331, 293, 311, 283, 241, 269, 349, 359, 383, 367, 401, 317, 379, 439, 491, 421, 409, 449, 463, 467

Offset: 1

Author

Amarnath Murthy, Sep 14 2002

Keywords

• nonn

Comments

Question: Is every prime a member of some pair?

If the distance between the prime pairs is not required to be 2n, we get A031215. - R. J. Mathar, Nov 26 2014

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

Examples

			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.
		

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A075322, A075323.

Programs

Haskell

a075321 = a075323 . subtract 1 . (* 2)
-- Reinhard Zumkeller, Nov 29 2014

Maple

A075321p := proc(n)
    option remember;
    local prevlist,i,p,q ;
    if n = 1 then
        return [3,5];
    else
        prevlist := [seq(op(procname(i)),i=1..n-1)] ;
        for i from 2 do
            p := ithprime(i) ;
            if not p in prevlist then
                q := p+2*n ;
                if isprime(q) and not q in prevlist then
                    return [p,q] ;
                end if;
            end if;
        end do:
    end if;
end proc:
A075321 := proc(n)
    op(1,A075321p(n)) ;
end proc:
seq(A075321(n),n=1..60) ; # R. J. Mathar, Nov 26 2014

Mathematica

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}]]]]];
A075321 [n_] := A075321p[n][[1]];
Array[A075321, 50] (* Jean-François Alcover, Feb 12 2018, translated from R. J. Mathar's program *)

Extensions

Corrected by R. J. Mathar, Nov 26 2014