cp's OEIS Frontend

A257250 Numbers n for which A256999(n) = n; numbers that cannot be made any larger by rotating (by one or more steps) the non-msb bits of their binary representation (with A080541 or A080542).

%I A257250 #64 Nov 05 2019 07:24:55
%S A257250 0,1,2,3,4,6,7,8,12,14,15,16,24,26,28,30,31,32,48,52,56,58,60,62,63,
%T A257250 64,96,100,104,106,112,114,116,118,120,122,124,126,127,128,192,200,
%U A257250 208,212,224,226,228,232,234,236,240,242,244,246,248,250,252,254,255,256,384,392,400,416,420,424,426,448,450
%N A257250 Numbers n for which A256999(n) = n; numbers that cannot be made any larger by rotating (by one or more steps) the non-msb bits of their binary representation (with A080541 or A080542).
%C A257250 These correspond to the maximal (lexicographically largest) representatives selected from each equivalence class of binary necklaces. See the last example.
%C A257250 Indexing starts from zero, because a(0) = 0 is a special case.
%C A257250 If k is a member then so also is 2*k, i.e., k with 0 appended to the end of its binary representation.
%C A257250 If k is a member then so also is A004755(k), i.e., k with 1 prepended to the front of its binary representation.
%C A257250 One obtains A065609 if one erases the most significant bit of each term [as A053645(a(n))] and then discards any zero-terms produced from the terms that originally were powers of two (A000079).
%C A257250 First differs from A328607 in lacking 108, with binary expansion 1101100. If we define a dual-necklace to be a finite sequence that is lexicographically maximal (not minimal) among all of its cyclic rotations, these are numbers whose binary expansion, without the most significant digit, is a dual-necklace. - _Gus Wiseman_, Nov 04 2019
%H A257250 Antti Karttunen, <a href="/A257250/b257250.txt">Table of n, a(n) for n = 0..16637</a>
%H A257250 Wikipedia, <a href="http://en.wikipedia.org/wiki/Necklace_%28combinatorics%29">Necklace (combinatorics)</a>
%H A257250 <a href="/index/Ne#necklaces">Index entries for sequences related to necklaces</a>
%e A257250 For n = 5, with binary representation "101", if we rotate other bits than the most significant bit (that is, only the two rightmost digits "01") one step to either direction, we get "110" = 6 > 5, so 5 can be made larger by such rotations, and thus is NOT included in this sequence.
%e A257250 For n = 6, with binary representation "110", no such rotation will yield a larger number, and thus 6 is included in this sequence.
%e A257250 For n = 28, with binary representation "11100", if we rotate non-msb bits towards right, we get additional numbers 22, 19 and 25 (with binary representations "10110", "10011", "11001") before coming to 28 again, and 28 is the largest of these numbers, thus 28 is included in this sequence.
%e A257250   Also, if we discard the most significant bit of each and consider them just as binary strings, then A053645(28) = 12 is the lexicographically largest representative of {"1100", "0110", "0011", "1001"}, which is the complete set of representatives for a particular equivalence class of binary necklaces, obtained by rotating all bits of binary string "1100" successively towards right or left.
%t A257250 reckQ[q_]:=Array[OrderedQ[{RotateRight[q,#],q}]&,Length[q]-1,1,And];
%t A257250 Select[Range[0,110],#<=1||reckQ[Rest[IntegerDigits[#,2]]]&] (* _Gus Wiseman_, Nov 04 2019 *)
%o A257250 (Scheme, with _Antti Karttunen_'s IntSeq-library)
%o A257250 (define A257250 (MATCHING-POS 0 0 (lambda (n) (= n (A256999 n)))))
%Y A257250 Complement: A257739.
%Y A257250 Odd terms: A000225.
%Y A257250 Subsequence of A065609.
%Y A257250 Cf. A004755, A053645, A080541, A080542, A256999.
%Y A257250 Subsequence: A258003.
%Y A257250 The non-dual version is A328668.
%Y A257250 The version involving all digits is A065609.
%Y A257250 The non-dual reversed version is A328607.
%Y A257250 Numbers whose reversed binary expansion is a necklace are A328595.
%Y A257250 Binary necklaces are A000031.
%Y A257250 Necklace compositions are A008965.
%Y A257250 Cf. A000120, A001037, A003714, A014081, A069010, A275692, A328594, A328596.
%K A257250 nonn,base
%O A257250 0,3
%A A257250 _Antti Karttunen_, May 16 2015