A326781 No position of a 1 in the reversed binary expansion of n is a power of 2.

0, 4, 16, 20, 32, 36, 48, 52, 64, 68, 80, 84, 96, 100, 112, 116, 256, 260, 272, 276, 288, 292, 304, 308, 320, 324, 336, 340, 352, 356, 368, 372, 512, 516, 528, 532, 544, 548, 560, 564, 576, 580, 592, 596, 608, 612, 624, 628, 768, 772, 784, 788, 800, 804, 816

Offset: 1

Author

Gus Wiseman, Jul 25 2019

Keywords

• nonn
• base

Comments

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:

0: {}

4: {{1,2}}

16: {{1,3}}

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

32: {{2,3}}

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

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

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

64: {{1,2,3}}

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

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

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

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

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

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

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

256: {{1,4}}

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

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

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

Examples

			The binary indices of n are row n of A048793. The sequence of terms together with their binary indices begins:
    0: {}
    4: {3}
   16: {5}
   20: {3,5}
   32: {6}
   36: {3,6}
   48: {5,6}
   52: {3,5,6}
   64: {7}
   68: {3,7}
   80: {5,7}
   84: {3,5,7}
   96: {6,7}
  100: {3,6,7}
  112: {5,6,7}
  116: {3,5,6,7}
  256: {9}
  260: {3,9}
  272: {5,9}
  276: {3,5,9}
		

Crossrefs

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

Programs

Mathematica

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
Select[Range[100],!MemberQ[Length/@bpe/@bpe[#],1]&]

Formula

Conjectures from Colin Barker, Jul 27 2019: (Start)

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)).

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

(End)