A203069 Lexicographically earliest sequence of distinct positive numbers such that a(n-1)+a(n) is odd and composite.

1, 8, 7, 2, 13, 12, 3, 6, 9, 16, 5, 4, 11, 10, 15, 18, 17, 22, 23, 26, 19, 14, 21, 24, 25, 20, 29, 28, 27, 30, 33, 32, 31, 34, 35, 40, 37, 38, 39, 36, 41, 44, 43, 42, 45, 46, 47, 48, 51, 54, 57, 58, 53, 52, 59, 56, 49, 50, 55, 60, 61, 62, 63, 66, 67, 68, 65

Offset: 1

Author

Zak Seidov, Dec 28 2011

Keywords

• nonn

Comments

Inspired by an idea of Eric Angelini on the Sequence Fans list on Dec 28 2011.

Comments from N. J. A. Sloane, Aug 16 2021: (Start)

It is conjectured that this is a permutation of the positive integers. Is there a proof? The terms are distinct, by definition, and the sequence is clearly infinite. But does every number appear?

In the first 100000 terms, the only differences a(i)-a(i-1) that occur are -11, -9, -7, -5, -3, -1, 1, 3, 5, 7, 9, 11 (see A346610).

Also a(n) is surprisingly close to n - see A346611. (End)

Examples

			a(1)=1; the smallest possible even number m such that 1+m is composite is m=8, hence a(2)=8;
the smallest possible odd number m such that 8+m is composite is m=7, hence a(3)=7;
the smallest possible even number m such that 7+m is composite is m=2, hence a(4)=2.
		

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
• Éric Angelini, A December quiz with a non-prime array, SeqFan list, Dec 28 2011.
• N. J. A. Sloane, Table of n, a(n) for n = 1..100000
• N. J. A. Sloane, Maple program
• Index entries for sequences that are permutations of the natural numbers

Crossrefs

Cf. A010051, A249918 (inverse), A014076, A055266, A346610 (first differences), A346611.

See A346609 for the successive odd nonprimes that arise.

Programs

Haskell

import Data.List (delete)
a203069 n = a203069_list !! (n-1)
a203069_list = 1 : f 1 [2..] where
   f u vs = g vs where
     g (w:ws) | odd z && a010051' z == 0 = w : f w (delete w vs)
              | otherwise = g ws
              where z = u + w
-- Reinhard Zumkeller, Jan 14 2015

Maple

(See link)

Mathematica

Clear[used];used={1};oc[n_]:=Module[{k=If[OddQ[n],2,1]},While[ !CompositeQ[ n+k]||MemberQ[used,k],k+=2];Flatten[AppendTo[used,k]];k] (* Harvey P. Dale, Aug 16 2021 *)

Sage

@cached_function
def A203069(n):
    if n == 1: return 1
    used = set(A203069(i) for i in [1..n-1])
    works = lambda an: (A203069(n-1)+an) % 2 == 1 and len(divisors((A203069(n-1)+an))) > 2
    return next(k for k in PositiveIntegers() if k not in used and works(k)) # D. S. McNeil, Dec 28 2011

Extensions

Revised by N. J. A. Sloane, Aug 15 2021 at the suggestion of Harvey P. Dale.