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A055265 a(n) is the smallest positive integer not already in the sequence such that a(n)+a(n-1) is prime, starting with a(1)=1. 39
1, 2, 3, 4, 7, 6, 5, 8, 9, 10, 13, 16, 15, 14, 17, 12, 11, 18, 19, 22, 21, 20, 23, 24, 29, 30, 31, 28, 25, 34, 27, 26, 33, 38, 35, 32, 39, 40, 43, 36, 37, 42, 41, 48, 49, 52, 45, 44, 53, 50, 47, 54, 55, 46, 51, 56, 57, 70, 61, 66, 65, 62, 69, 58, 73, 64, 63, 68, 59, 72, 67, 60 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

The sequence is well-defined (the terms must alternate in parity, and by Dirichlet's theorem a(n+1) always exists). - N. J. A. Sloane, Mar 07 2017

Does every positive integer eventually occur? - Dmitry Kamenetsky, May 27 2009. Reply from Robert G. Wilson v, May 27 2009: The answer is almost certainly yes, on probabilistic grounds.

It appears that this is the limit of the rows of A051237. That those rows do approach a limit seems certain, and given that that limit exists, that this sequence is the limit seems even more likely, but no proof is known for either conjecture. - Robert G. Wilson v, Mar 11 2011, edited by Franklin T. Adams-Watters, Mar 17 2011

The sequence is also a particular case of "among the pairwise sums of any M consecutive terms, N are prime", with M = 2, N = 1. For other M, N see A055266 & A253074 (M = 2, N = 0), A329333, A329405 - A329416, A329449 - A329456, A329563 - A329581, and the OEIS Wiki page. - M. F. Hasler, Feb 11 2020

LINKS

Zak Seidov, Table of n, a(n) for n = 1..10000 (First 1000 terms from T. D. Noe)

N. J. A. Sloane, Table of n, a(n) for n = 1..100000 (computed using Orlovsky's Mma program)

M. F. Hasler, Prime sums from neighboring terms, OEIS Wiki, Nov. 23, 2019

Index entries for sequences that are permutations of the natural numbers

FORMULA

a(2n-1) = A128280(2n-1) - 1, a(2n) = A128280(2n) + 1, for all n >= 1. - M. F. Hasler, Feb 11 2020

EXAMPLE

a(5) = 7 because 1, 2, 3 and 4 have already been used and neither 4 + 5 = 9 nor 4 + 6 = 10 are prime while 4 + 7 = 11 is prime.

MAPLE

A055265 := proc(n)

    local a, i, known ;

    option remember;

    if n =1 then

        1;

    else

        for a from 1 do

            known := false;

            for i from 1 to n-1 do

                if procname(i) = a then

                    known := true;

                    break;

                end if;

            end do:

            if not known and isprime(procname(n-1)+a) then

                return a;

            end if;

        end do:

    end if;

end proc:

seq(A055265(n), n=1..100) ; # R. J. Mathar, Feb 25 2017

MATHEMATICA

f[s_List] := Block[{k = 1, a = s[[ -1]]}, While[ MemberQ[s, k] || ! PrimeQ[a + k], k++ ]; Append[s, k]]; Nest[f, {1}, 71] (* Robert G. Wilson v, May 27 2009 *)

q=2000; a={1}; z=Range[2, 2*q]; While[Length[z]>q-1, k=1; While[!PrimeQ[z[[k]]+Last[a]], k++]; AppendTo[a, z[[k]]]; z=Delete[z, k]]; Print[a] (*200 times faster*) (* Vladimir Joseph Stephan Orlovsky, May 03 2011 *)

PROG

(HP 50G Calculator) << DUPDUP + 2 -> N M L << { 1 } 1 N 1 - FOR i L M FOR j DUP j POS NOT IF THEN j DUP 'L' STO M 'j' STO END NEXT OVER i GET SWAP WHILE DUP2 + DUP ISPRIME? NOT REPEAT DROP DO 1 + 3 PICK OVER POS NOT UNTIL END END ROT DROP2 + NEXT >> >> Gerald Hillier, Oct 28 2008

(Haskell)

import Data.List (delete)

a055265 n = a055265_list !! (n-1)

a055265_list = 1 : f 1 [2..] where

   f x vs = g vs where

     g (w:ws) = if a010051 (x + w) == 1

                   then w : f w (delete w vs) else g ws

-- Reinhard Zumkeller, Feb 14 2013

(PARI) v=[1]; n=1; while(n<50, if(isprime(v[#v]+n)&&!vecsearch(vecsort(v), n), v=concat(v, n); n=0); n++); v \\ Derek Orr, Jun 01 2015

(PARI) U=-a=1; vector(100, k, k=valuation(1+U+=1<<a, 2); while(bittest(U, k)|| !isprime(a+k), k++); a=k) \\ M. F. Hasler, Feb 11 2020

CROSSREFS

Inverse permutation: A117922; fixed points: A117925; A117923=a(a(n)). - Reinhard Zumkeller, Apr 03 2006

Cf. A036440, A051237, A051239, A055266, A088643. A010051.

Cf. A086527 (the primes a(n)+a(n-1)).

Cf. A070942 (n's such that a(1..n) is a permutation of (1..n)). - Zak Seidov, Oct 19 2011

See also A076990, A243625.

See A282695 for deviation from identity sequence.

A073659 is a version where the partial sums must be primes.

Sequence in context: A264965 A266644 A329411 * A117922 A266643 A321525

Adjacent sequences:  A055262 A055263 A055264 * A055266 A055267 A055268

KEYWORD

easy,nice,nonn

AUTHOR

Henry Bottomley, May 09 2000

EXTENSIONS

Corrected by Hans Havermann, Sep 24 2002

STATUS

approved

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Last modified May 27 03:03 EDT 2020. Contains 334647 sequences. (Running on oeis4.)