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A374356 a(n) is the greatest fibbinary number f <= n such that n - f is also a fibbinary number whose binary expansion has no common 1's with that of f (where fibbinary numbers correspond to A003714).

%I A374356 #16 Jul 13 2025 11:06:42
%S A374356 0,1,2,2,4,5,4,5,8,9,10,10,8,9,10,10,16,17,18,18,20,21,20,21,16,17,18,
%T A374356 18,20,21,20,21,32,33,34,34,36,37,36,37,40,41,42,42,40,41,42,42,32,33,
%U A374356 34,34,36,37,36,37,40,41,42,42,40,41,42,42,64,65,66,66
%N A374356 a(n) is the greatest fibbinary number f <= n such that n - f is also  a fibbinary number whose binary expansion has no common 1's with that of f (where fibbinary numbers correspond to A003714).
%C A374356 To compute a(n): replace every other bit with zero (starting with the second bit) in each run of consecutive 1's in the binary expansion of n.
%C A374356 From _Gus Wiseman_, Jul 11 2025: (Start)
%C A374356 This is the greatest binary rank of a sparse subset of the binary indices of n, where:
%C A374356 1. The binary indices of a nonnegative integer are the positions of 1 in its reversed binary expansion.
%C A374356 2. A set is sparse iff 1 is not a first difference.
%C A374356 3. The binary rank of a set {S_1,S_2,...} is Sum_i 2^(S_i-1).
%C A374356 (End)
%H A374356 Rémy Sigrist, <a href="/A374356/b374356.txt">Table of n, a(n) for n = 0..8191</a>
%H A374356 <a href="/index/Bi#binary">Index entries for sequences related to binary expansion of n</a>
%F A374356 a(n) = A374354(n, A277561(n)-1).
%F A374356 a(n) = n - A374355(n).
%F A374356 a(n) <= n with equality iff n is a fibbinary number.
%e A374356 The first terms, in decimal and in binary, are:
%e A374356   n   a(n)  bin(n)  bin(a(n))
%e A374356   --  ----  ------  ---------
%e A374356    0     0       0          0
%e A374356    1     1       1          1
%e A374356    2     2      10         10
%e A374356    3     2      11         10
%e A374356    4     4     100        100
%e A374356    5     5     101        101
%e A374356    6     4     110        100
%e A374356    7     5     111        101
%e A374356    8     8    1000       1000
%e A374356    9     9    1001       1001
%e A374356   10    10    1010       1010
%e A374356   11    10    1011       1010
%e A374356   12     8    1100       1000
%e A374356   13     9    1101       1001
%e A374356   14    10    1110       1010
%e A374356   15    10    1111       1010
%e A374356   16    16   10000      10000
%t A374356 bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
%t A374356 fbi[q_]:=If[q=={},0,Total[2^q]/2];
%t A374356 Table[Max@@fbi/@Select[Subsets[bpe[n]],FreeQ[Differences[#],1]&],{n,0,100}] (* _Gus Wiseman_, Jul 11 2025 *)
%o A374356 (PARI) a(n) = { my (v = 0, e, x, y, b); while (n, x = y = 0; e = valuation(n, 2); for (k = 0, oo, if (bittest(n, e+k), n -= b = 2^(e+k); [x, y] = [y + b, x], v += x; break;););); return (v); }
%Y A374356 Cf. A277561, A374354, A374355.
%Y A374356 The union is A003714 (Fibbinary numbers).
%Y A374356 For prime instead of binary indices we have A385216.
%Y A374356 A034839 counts subsets by number of maximal runs, for strict partitions A116674.
%Y A374356 A166469 counts sparse submultisets of prime indices, maximal A385215.
%Y A374356 A245564 counts sparse subsets of binary indices, maximal case A384883.
%Y A374356 A319630 ranks sparse submultisets of prime indices, complement A104210.
%Y A374356 Cf. A000045, A000071, A001629, A006519, A010049, A044813, A119900, A202023, A202064, A268193, A384890.
%K A374356 nonn,base
%O A374356 0,3
%A A374356 _Rémy Sigrist_, Jul 06 2024