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.

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

Offset: 1

Author

Robert G. Wilson v, Oct 24 2017

Keywords

• nonn
• base

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

Crossrefs

Cf. A000040, A000079, A059459, A094076.

Programs

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]