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A209554 Primes that expressed in none of the forms n<+>2 and n<+>3, where the operation <+> is defined in A206853. 6
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

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.

LINKS

Table of n, a(n) for n=1..45.

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 *)

CROSSREFS

Cf. A209544.

Sequence in context: A030262 A107173 A100494 * A243155 A201843 A278202

Adjacent sequences:  A209551 A209552 A209553 * A209555 A209556 A209557

KEYWORD

nonn,base

AUTHOR

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

STATUS

approved

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Last modified September 22 09:19 EDT 2017. Contains 292336 sequences.