%I #12 Jul 27 2019 14:51:27

%S 0,4,16,20,32,36,48,52,64,68,80,84,96,100,112,116,256,260,272,276,288,

%T 292,304,308,320,324,336,340,352,356,368,372,512,516,528,532,544,548,

%U 560,564,576,580,592,596,608,612,624,628,768,772,784,788,800,804,816

%N No position of a 1 in the reversed binary expansion of n is a power of 2.

%C Also BII-numbers (see A326031) of set-systems with no singleton edges. For example, the sequence of such set-systems together with their BII-numbers begins:

%C 0: {}

%C 4: {{1,2}}

%C 16: {{1,3}}

%C 20: {{1,2},{1,3}}

%C 32: {{2,3}}

%C 36: {{1,2},{2,3}}

%C 48: {{1,3},{2,3}}

%C 52: {{1,2},{1,3},{2,3}}

%C 64: {{1,2,3}}

%C 68: {{1,2},{1,2,3}}

%C 80: {{1,3},{1,2,3}}

%C 84: {{1,2},{1,3},{1,2,3}}

%C 96: {{2,3},{1,2,3}}

%C 100: {{1,2},{2,3},{1,2,3}}

%C 112: {{1,3},{2,3},{1,2,3}}

%C 116: {{1,2},{1,3},{2,3},{1,2,3}}

%C 256: {{1,4}}

%C 260: {{1,2},{1,4}}

%C 272: {{1,3},{1,4}}

%C 276: {{1,2},{1,3},{1,4}}

%F Conjectures from _Colin Barker_, Jul 27 2019: (Start)

%F G.f.: 4*x^2*(1 + 3*x + x^2 + 3*x^3 + x^4 + 3*x^5 + x^6 + 3*x^7 + x^8 + 3*x^9 + x^10 + 3*x^11 + x^12 + 3*x^13 + x^14 + 35*x^15) / ((1 - x)^2*(1 + x)*(1 + x^2)*(1 + x^4)*(1 + x^8)).

%F a(n) = a(n-1) + a(n-16) - a(n-17) for n>17.

%F (End)

%e The binary indices of n are row n of A048793. The sequence of terms together with their binary indices begins:

%e 0: {}

%e 4: {3}

%e 16: {5}

%e 20: {3,5}

%e 32: {6}

%e 36: {3,6}

%e 48: {5,6}

%e 52: {3,5,6}

%e 64: {7}

%e 68: {3,7}

%e 80: {5,7}

%e 84: {3,5,7}

%e 96: {6,7}

%e 100: {3,6,7}

%e 112: {5,6,7}

%e 116: {3,5,6,7}

%e 256: {9}

%e 260: {3,9}

%e 272: {5,9}

%e 276: {3,5,9}

%t bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];

%t Select[Range[100],!MemberQ[Length/@bpe/@bpe[#],1]&]

%Y Cf. A000120, A029931, A048793, A062289, A070939, A326031, A326782, A326788.

%K nonn,base

%O 1,2

%A _Gus Wiseman_, Jul 25 2019