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A203069 Lexicographically earliest sequence of distinct positive numbers such that a(n-1)+a(n) is odd and composite. 8
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

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)

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

EXAMPLE

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.

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 *)

PROG

(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

(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

CROSSREFS

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

See A346609 for the successive odd nonprimes that arise.

Sequence in context: A155068 A244839 A329450 * A343626 A272531 A244684

Adjacent sequences: A203066 A203067 A203068 * A203070 A203071 A203072

KEYWORD

nonn

AUTHOR

Zak Seidov, Dec 28 2011

EXTENSIONS

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

STATUS

approved

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Last modified December 5 21:40 EST 2022. Contains 358594 sequences. (Running on oeis4.)