A087086 Primitive sets of integers, each subset mapped onto a unique binary integer, values here shown in decimal.

0, 1, 2, 4, 6, 8, 12, 16, 18, 20, 22, 24, 28, 32, 40, 48, 56, 64, 66, 68, 70, 72, 76, 80, 82, 84, 86, 88, 92, 96, 104, 112, 120, 128, 132, 144, 148, 160, 176, 192, 196, 208, 212, 224, 240, 256, 258, 264, 272, 274, 280, 288, 296, 304, 312, 320, 322, 328, 336, 338, 344

Offset: 0

Author

Alan Sutcliffe (alansut(AT)ntlworld.com), Aug 14 2003

Keywords

• easy
• nonn
• base
• changed

Comments

A primitive set of integers has no pair of elements one of which divides the other. Each element i in a subset contributes 2^(i-1) to the binary value for that subset. The integers missing from the sequence correspond to nonprimitive subsets.

Examples

			a(10)=22 since the 10th primitive set counting from 0 is {5,3,2}, which maps onto 10110 binary = 22 decimal.
From _Gus Wiseman_, Oct 31 2019: (Start)
The sequence of terms together with their binary expansions and binary indices begins:
   0:       0 ~ {}
   1:       1 ~ {1}
   2:      10 ~ {2}
   4:     100 ~ {3}
   6:     110 ~ {2,3}
   8:    1000 ~ {4}
  12:    1100 ~ {3,4}
  16:   10000 ~ {5}
  18:   10010 ~ {2,5}
  20:   10100 ~ {3,5}
  22:   10110 ~ {2,3,5}
  24:   11000 ~ {4,5}
  28:   11100 ~ {3,4,5}
(End)
		

References

• Alan Sutcliffe, Divisors and Common Factors in Sets of Integers, awaiting publication

Crossrefs

A051026 gives the number of primitive subsets of the integers 1 to n.

The version for prime indices (rather than binary indices) is A316476.

The relatively prime case is A328671.

Partitions with no consecutive divisible parts are A328171.

Compositions without consecutive divisible parts are A328460.

A ranking of antichains is A326704.

Cf. A000120, A048793, A070939, A285572, A285573, A303362, A304713, A305148, A328593.

Programs

Mathematica

stableQ[u_,Q_]:=!Apply[Or,Outer[#1=!=#2&&Q[#1,#2]&,u,u,1],{0,1}];
Select[Range[0,100],stableQ[Join@@Position[Reverse[IntegerDigits[#,2]],1],Divisible]&] (* Gus Wiseman, Oct 31 2019 *)