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%I #27 May 28 2018 12:19:59
%S 2,3,5,7,11,13,29,17,19,23,31,47,37,41,43,59,61,53,101,71,67,73,79,
%T 103,97,107,109,127,191,151,131,137,139,163,167,173,157,149,181,179,
%U 211,83,89,113,241,193,197,199,223,239,227,229,233,251,379,283,271,263,257,269,277,281,313,307,311,293
%N Begin with 2; thereafter a(n) is the least prime not already in the sequence such that the Hamming distance between it and the preceding prime is at most 2.
%C The Hamming distance between two primes p and q is the Hamming distance between their binary expansions. - _N. J. A. Sloane_, May 27 2018
%C Conjecture: this sequence is a permutation of the primes.
%C By definition, the absolute difference of a(n) and a(n + 1) is in A048645. - _David A. Corneth_, Jan 12 2018
%H Robert G. Wilson v, <a href="/A294994/b294994.txt">Table of n, a(n) for n = 1..10000</a>
%H Rémy Sigrist, <a href="/A294994/a294994.png">Scatterplot of (x, y) such that the prime numbers prime(x) and prime(y) have Hamming distance <= 2 for x = 1..1000 and y = 1..1000</a>
%t f[s_List] := Block[{p = s[[-1]], q = 3}, While[MemberQ[s, q] || Plus @@ IntegerDigits [BitXor[p, q], 2] > 2, q = NextPrime@q]; Append[s, q]]; s = {2}; Nest[f, s, 65]
%o (PARI) s = 0; v = 2; for (n=1, 66, print1 (v ", "); s += 2^v; forprime (p=2, oo, if (!bittest(s, p) && hammingweight(bitxor(p, v))<=2, v = p; break))) \\ _Rémy Sigrist_, Jan 08 2018
%Y Cf. A000040, A004676, A048645, A163252.
%Y See also A303593, A303594, A303595 (when n-th prime appears).
%K nonn,base
%O 1,1
%A _Robert G. Wilson v_, Nov 12 2017