A055266 a(n) + a(n+1) is never prime; lexicographically earliest such sequence of distinct positive integers.

1, 3, 5, 4, 2, 6, 8, 7, 9, 11, 10, 12, 13, 14, 16, 17, 15, 18, 20, 19, 21, 23, 22, 24, 25, 26, 28, 27, 29, 31, 32, 30, 33, 35, 34, 36, 38, 37, 39, 41, 40, 42, 43, 44, 46, 45, 47, 48, 50, 49, 51, 53, 52, 54, 56, 55, 57, 58, 59, 60, 61, 62, 63, 65, 64, 66, 67, 68, 70, 71, 69, 72

Offset: 1

Author

Henry Bottomley, May 09 2000

Keywords

• easy
• nonn

Comments

See A253074 for an essentially identical sequence (with a proof that the sequence is a permutation).

Sequence A253074 is defined in the same way, but starting with 0. This happens to produce the same sequence from the next term on. This is the case (M,N) = (2,0) in the family of sequences where M consecutive terms yield N primes in their pairwise sums, see the wiki page for other examples. - M. F. Hasler, Nov 26 2019

Examples

			a(3) = 5 because 1 and 3 have already been used and both 3 + 2 = 5 and 3 + 4 = 7 are prime while 3 + 5 = 8 is not prime.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000
• 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

Cf. A055265, A010051, A249920 (inverse), A203069, A253074.

Programs

Haskell

import Data.List (delete)
a055266 n = a055266_list !! (n-1)
a055266_list = 1 : f 1 [2..] where
   f u vs = g vs where
     g (w:ws) | a010051' (u + w) == 0 = w : f w (delete w vs)
              | otherwise = g ws
-- Reinhard Zumkeller, Jan 14 2015

Maple

N:= 1000; # to get a[n] for n up to N
A:= {1};
a[1]:= 1;
for n from 2 to N do
  mA:= max(A);
  R:= {$1..mA} minus A;
  for x in R do
     if not isprime(a[n-1]+x) then
       a[n]:= x;
       break
     fi
   od:
   if not assigned(a[n]) then
     for x from mA+1 do
       if not isprime(a[n-1]+x) then
         a[n]:= x;
         break
       fi
     od
   fi;
   A:= A union {x};
od:
seq(a[n],n=1..N); # Robert Israel, Jun 03 2014

Mathematica

f[ s_ ]:=Block[ {k=1,a=s[ [ -1 ] ]},While[ Or[ MemberQ[ s,k ],PrimeQ[ a+k ] ],k++ ];Append[ s,k ] ];Nest[ f,{1},121 ] (* Zak Seidov, Oct 21 2009 *)
a={1};z=Range[2,2002];z=Complement[z,a];While[Length[z]>1,If[!PrimeQ[z[[1]]+Last[a]],AppendTo[a,z[[1]]],If[!PrimeQ[z[[2]]+Last[a]],AppendTo[a,z[[2]]],AppendTo[a,z[[3]]]]];z=Complement[z,a]];Print[a] (* significantly faster *) (* Vladimir Joseph Stephan Orlovsky, May 03 2011 *)

PARI

v=[1]; n=1; while(n<100, if(!isprime(n+v[#v])&&!vecsearch(vecsort(v), n), v=concat(v, n); n=0); n++); v \\ Derek Orr, Jun 08 2015

PARI

A055266_upto(n=99, u=1, U, a)={vector(n, n, n=u; while(bittest(U, n-u)|| isprime(a+n), n++); if(n>u, U+=1<<(n-u), U>>=-u+u+=valuation(U+2, 2)); a=n) + if(default(debug), print([u]))} \\ Optional args allow to tweak computation. If debug > 0, print least unused number at the end. - M. F. Hasler, Nov 25 2019

Formula

a(n) = A253074(n+1) (as long as A253074(1) = 0). - M. F. Hasler, Nov 26 2019

Extensions

Corrected by Zak Seidov, Oct 21 2009

Name edited by M. F. Hasler, Nov 26 2019