|
| |
|
|
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
|
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.
|
|
|
LINKS
|
|
|
|
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
|
|
|
|
KEYWORD
|
nonn,base
|
|
|
AUTHOR
|
|
|
|
STATUS
|
approved
|
| |
|
|
