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.

3, 5, 7, 11, 13, 19, 23, 31, 37, 47, 17, 29, 53, 67, 43, 59, 61, 79, 83, 103, 109, 131, 73, 97, 101, 127, 139, 167, 41, 71, 149, 181, 157, 191, 137, 173, 113, 151, 193, 233, 197, 239, 179, 223, 211, 257, 229, 277, 263, 313, 199

Offset

1,1

Comments

Question: Is every odd prime a member of some pair?

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

Links

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

Maple

# A075321p implemented in A075321
A075323 := proc(n)
    if type(n, 'odd') then
        op(1, A075321p((n+1)/2)) ;
    else
        op(2, A075321p(n/2)) ;
    end if;
end proc:
seq(A075323(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}]]]]];
A075323[n_] := If[OddQ[n], A075321p[(n+1)/2][[1]], A075321p[n/2][[2]]];
Array[A075323, 50] (* Jean-François Alcover, Feb 12 2018, after R. J. Mathar *)

Prog

(Haskell)
import Data.List ((\\))
a075323 n = a075323_list !! (n-1)
a075323_list = f 1 []  $ tail a000040_list where
   f k ys qs = g qs where
     g (p:ps) | a010051' pk == 0 || pk `elem` ys = g ps
              | otherwise = p : pk : f (k + 1) (p:pk:ys) (qs \\ [p, pk])
              where pk = p + 2 * k
-- Reinhard Zumkeller, Nov 29 2014

Crossrefs

Cf. A075321, A075322.

Cf. A010051, A000040, A065091, A247233.

Sequence in context: A132779 A192869 A147513 * A020575 A055072 A059334

Adjacent sequences: A075320 A075321 A075322 * A075324 A075325 A075326

Keyword

nonn

Author

Amarnath Murthy, Sep 14 2002

Extensions

Corrected by R. J. Mathar, Nov 26 2014

Status

approved