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

1,1

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).

Links

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

Example

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.

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 *)

Prog

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

Crossrefs

Cf. A075322, A075323.

Sequence in context: A101301 A103116 A303853 * A258030 A164787 A131205

Adjacent sequences: A075318 A075319 A075320 * A075322 A075323 A075324

Keyword

nonn

Author

Amarnath Murthy, Sep 14 2002

Extensions

Corrected by R. J. Mathar, Nov 26 2014

Status

approved