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%I #50 Jul 10 2021 04:20:35
%S 3,97,193,257,353,449,577,641,673,769,929,1153,1217,1249,1409,1601,
%T 1697,1889,2017,2081,2113,2273,2593,2657,2689,2753,3041,3137,3169,
%U 3329,3361,3457,3617,4001,4129,4289,4481,4513,4673,4801,4993,5153,5281,5441,5569
%N Primes that expressed in none of the forms n<+>2 and n<+>3, where the operation <+> is defined in A206853.
%C How relate these to A133870? - _R. J. Mathar_, Mar 13 2012
%C If the formulated below conjecture is true, then for n>=2, A209544 and this sequence coincide with A007519 and A133870 respectively.
%C Let n>=3 be odd and k>=2. We say that n possesses a property S_k, if for every integer m from interval [0,n) with the Hamming distance D(m,n) in [2,k], there exists an integer h from (m,n) with D(m,h)=D(m,n).
%C Conjecture (A209544 and this sequence correspond to cases k=2 and k=3 respectively).
%C Odd n>3 possesses the property S_k iff n has the form n=2^(2*k-1)*t+1.
%C Example. Let k=2, t=1. Then n=9=(1001)_2. All numbers m from [0,9) with D(m,9)=2 are 0,3,5.
%C For m=0, we can take h=3, since 3 from (0,9) and D(0,3)=2; for m=3, we can take h=5, since 5 from (3,9) and D(3,5)=2; for m=5, we can take h=6, since 6 from (5,9) and D(5,6)=2. - _Vladimir Shevelev_, Seqfan list Apr 05 2012.
%C For k=2 this conjecture is true (see comment in A182187). - _Vladimir Shevelev_, Apr 18 2012.
%t hammingDistance[a_,b_] := Count[IntegerDigits[BitXor[a,b],2],1]; vS[a_,b_] := NestWhile[#+1&,a,hammingDistance[a,#]=!=b&]; (* vS[a_,b_] is the least c>=a, such that the binary Hamming distance D (a,c)=b.vS[a,b] is Vladimir's a<+>b *) A209554 = Apply[Intersection, Table[Map[Prime[#]&, Complement[Range[Last[#]], #]&[Map[PrimePi[#]&, Union[Map[#[[2]]&, Cases[Map[{PrimeQ[#], #}&[vS[#,n]]&, Range[7500]], {True,_}]]]]]],{n, 2, 3}]] (* be careful with ranges near 2^x *)
%Y Cf. A209544, A182187 (n<+>2), A182209 (n<+>3).
%Y Cf. A133870.
%K nonn,base
%O 1,1
%A _Vladimir Shevelev_ and _Peter J. C. Moses_, Mar 10 2012
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