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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 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<

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