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.

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

Offset: 1

Author

Henry Bottomley, May 09 2000

Keywords

• easy
• nice
• nonn

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

Examples

			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.
		

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

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.

Programs

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

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 *)

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<M. F. Hasler, Feb 11 2020

Formula

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

Extensions

Corrected by Hans Havermann, Sep 24 2002