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 A294205 a(1) = 2; for n > 1, a(n) is the least prime p not already in the sequence such that the Hamming distance between p and a(n-1) is 1. 1
 2, 3, 7, 5, 13, 29, 31, 23, 19, 17, 8209, 8273, 10321, 2129, 2113, 3137, 3169, 19553, 19489, 19457, 18433, 83969, 84481, 84737, 2181889, 2181953, 2706241, 2704193, 2687809, 590657, 590593, 590609, 590641, 590129, 524593, 274878431537, 274878431521, 274879480097, 1573153, 1573217, 1704289, 5898593 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS Conjecture: this sequence is infinite. First differs from A059459 at a(15). By definition of a Hamming distance of 1, the first forward absolute difference between a(n-1) and a(n) is a power of two (A000079). The exponent of two in those differences is 0, 2, 1, 3, 4, 1, 3, 2, 1, 13, 6, 11, 13, 4, 10, 5, 14, 6, 5, 10, 16, 9, 8, 21, 6, 19, 11, 14, 21, 6, 4, 5, 9, 16, 38, 4, 20, 38, 6, 17, 22, 20, 14, 22, 10, 14, 2, 10, 46, 1, 28, 3, 56, 75, 3, 8, 16, 27, 75, 3, 20, 25, 606, 807, 2052, 2177, 886, 759, 796, 5357, 966, 399, etc. Note that it is not true that for every prime m there is some k such that m+2^k is prime: see comments and links at A094076. Thus it is quite conceivable that the sequence is finite. - Robert Israel, Nov 15 2017 LINKS Robert G. Wilson v, Table of n, a(n) for n = 1..70 MAPLE A[1]:= 2: S:= {2}: L:= [1]: for n from 2 to 50 do   found:= false;   for i from 1 to nops(L) while not found do     cand:= A[n-1] - 2^L[-i];     if not member(cand, S) and isprime(cand) then       found:= true; L:= subsop(-i=NULL, L) fi;   od;   for k from 0 while not found do     if not member(k, L) then       cand:= A[n-1] + 2^k;       if not member(cand, S) and isprime(cand) then         found:= true; L:= sort([op(L), k]);       fi     fi   od;   A[n]:= cand;   S:= S union {cand}; od: seq(A[i], i=1..50); # Robert Israel, Nov 15 2017 MATHEMATICA hammingDistance[a_, b_] := Count[ IntegerDigits[ BitXor[a, b], 2], 1]; f[s_List] := Block[{p = s[[-1]], q = 3}, While[MemberQ[s, q] || hammingDistance[p, q] > 1, q = NextPrime@q]; Append[s, q]]; s = {2}; Nest[f, s, 26] (* or *) f[s_List] := Block[{k = -Floor[RealExponent[s[[-1]], 2]], p = s[[-1]]}, While[q = If[k < 0, p - 2^-k, p + 2^k]; MemberQ[s, q] || !PrimeQ[q] || hammingDistance[p, q] > 1, k++]; Append[s, q]]; s = {2}; Nest[f, s, 67] CROSSREFS Cf. A000040, A000079, A059459, A094076. Sequence in context: A064011 A050367 A192175 * A059459 A124440 A067363 Adjacent sequences:  A294202 A294203 A294204 * A294206 A294207 A294208 KEYWORD nonn,base AUTHOR Robert G. Wilson v, Oct 24 2017 STATUS approved

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Last modified October 14 17:31 EDT 2019. Contains 328022 sequences. (Running on oeis4.)