A075322 -id:A075322 - cp's OEIS frontend

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

Amarnath Murthy, Sep 14 2002

nonn

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

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

Cf. A075321, A075322.

Cf. A010051, A000040, A065091, A247233.

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

# 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

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 *)

Corrected by R. J. Mathar, Nov 26 2014

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.

Original entry on oeis.org

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

Amarnath Murthy, Sep 14 2002

nonn

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).

			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.

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

Cf. A075322, A075323.

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

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

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 *)

Corrected by R. J. Mathar, Nov 26 2014

cp's OEIS Frontend

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.

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