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A257311 a(1) = 4; a(2) = 5; for n > 2, a(n) is the smallest number of the form prime + 2 not already used which shares a factor with a(n-1).

%I A257311 #17 Apr 24 2015 10:55:52
%S A257311 4,5,15,9,21,7,49,63,33,39,13,91,105,25,45,55,75,69,81,99,111,129,43,
%T A257311 559,169,195,85,115,165,141,153,159,183,61,549,201,213,225,175,133,19,
%U A257311 285,231,243,273,259,315,235,265,295,355,375,279,31,403,351,309,103
%N A257311 a(1) = 4; a(2) = 5; for n > 2, a(n) is the smallest number of the form prime + 2 not already used which shares a factor with a(n-1).
%C A257311 Analog of EKG-sequence (A064413) on the numbers of the form prime + 2.
%C A257311 Conjecture: the sequence {a(n)-2} is a permutation of the primes (A000040).
%C A257311 Every prime in the sequence is greater of twin primes (A006512).
%C A257311 A generalization. Let A_k (k>=1) be the following sequence: a(1) = 2^k+2; a(2) = 2^k+3; for n > 2, a(n) is the smallest number of the form 2^k+prime not already used which shares a factor with a(n-1).
%C A257311 Conjecture: For every k>=1, the sequence A_k - 2^k is a permutation of the primes.
%C A257311 A_1 = A257311, A_2 = A257312, A_3 = A257313, A_4 = A257314, A_5 = A257315.
%H A257311 Peter J. C. Moses, <a href="/A257311/b257311.txt">Table of n, a(n) for n = 1..1000</a>
%t A257311 f[n_] := Block[{o = 2, s, p, k}, s = {o + 2, o + 3}; For[k = 3, k <= n, k++, p = 2; While[GCD[p + o, s[[k - 1]]] == 1 || MemberQ[s, p + o], p = NextPrime@ p]; AppendTo[s, p + o]]; s]; f@ 58 (* _Michael De Vlieger_, Apr 20 2015 *)
%Y A257311 Cf. A000040, A006512, A064413, A257312, A257313, A257314, A257315.
%K A257311 nonn
%O A257311 1,1
%A A257311 _Vladimir Shevelev_, Apr 20 2015
%E A257311 More terms from _Peter J. C. Moses_, Apr 20 2015