A055266 - OEIS

%I #39 Jan 24 2021 10:29:57

%S 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,

%T 28,27,29,31,32,30,33,35,34,36,38,37,39,41,40,42,43,44,46,45,47,48,50,

%U 49,51,53,52,54,56,55,57,58,59,60,61,62,63,65,64,66,67,68,70,71,69,72

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

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

%C 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

%H Robert Israel, <a href="/A055266/b055266.txt">Table of n, a(n) for n = 1..10000</a>

%H M. F. Hasler, <a href="/wiki/User:M._F._Hasler/Prime_sums_from_neighboring_terms">Prime sums from neighboring terms</a>, OEIS wiki, Nov. 23, 2019

%H <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>

%F a(n) = A253074(n+1) (as long as A253074(1) = 0). - _M. F. Hasler_, Nov 26 2019

%e 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.

%p N:= 1000; # to get a[n] for n up to N

%p A:= {1};

%p a[1]:= 1;

%p for n from 2 to N do

%p   mA:= max(A);

%p   R:= {$1..mA} minus A;

%p   for x in R do

%p      if not isprime(a[n-1]+x) then

%p        a[n]:= x;

%p        break

%p      fi

%p    od:

%p    if not assigned(a[n]) then

%p      for x from mA+1 do

%p        if not isprime(a[n-1]+x) then

%p          a[n]:= x;

%p          break

%p        fi

%p      od

%p    fi;

%p    A:= A union {x};

%p od:

%p seq(a[n],n=1..N); # _Robert Israel_, Jun 03 2014

%t 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 *)

%t 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 *)

%o (Haskell)

%o import Data.List (delete)

%o a055266 n = a055266_list !! (n-1)

%o a055266_list = 1 : f 1 [2..] where

%o    f u vs = g vs where

%o      g (w:ws) | a010051' (u + w) == 0 = w : f w (delete w vs)

%o               | otherwise = g ws

%o -- _Reinhard Zumkeller_, Jan 14 2015

%o (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

%o (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

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

%K easy,nonn

%O 1,2

%A _Henry Bottomley_, May 09 2000

%E Corrected by _Zak Seidov_, Oct 21 2009

%E Name edited by _M. F. Hasler_, Nov 26 2019