A209554 Primes that expressed in none of the forms n<+>2 and n<+>3, where the operation <+> is defined in A206853.

3, 97, 193, 257, 353, 449, 577, 641, 673, 769, 929, 1153, 1217, 1249, 1409, 1601, 1697, 1889, 2017, 2081, 2113, 2273, 2593, 2657, 2689, 2753, 3041, 3137, 3169, 3329, 3361, 3457, 3617, 4001, 4129, 4289, 4481, 4513, 4673, 4801, 4993, 5153, 5281, 5441, 5569

Offset: 1

Author

Vladimir Shevelev and Peter J. C. Moses, Mar 10 2012

Keywords

• nonn
• base

Comments

How relate these to A133870? - R. J. Mathar, Mar 13 2012

If the formulated below conjecture is true, then for n>=2, A209544 and this sequence coincide with A007519 and A133870 respectively.

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).

Conjecture (A209544 and this sequence correspond to cases k=2 and k=3 respectively).

Odd n>3 possesses the property S_k iff n has the form n=2^(2*k-1)*t+1.

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.

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.

For k=2 this conjecture is true (see comment in A182187). - Vladimir Shevelev, Apr 18 2012.

Crossrefs

Cf. A209544, A182187 (n<+>2), A182209 (n<+>3).

Cf. A133870.

Programs

Mathematica

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 *)