A096827 Number of antichains in divisor lattice D(n).

2, 3, 3, 4, 3, 6, 3, 5, 4, 6, 3, 10, 3, 6, 6, 6, 3, 10, 3, 10, 6, 6, 3, 15, 4, 6, 5, 10, 3, 20, 3, 7, 6, 6, 6, 20, 3, 6, 6, 15, 3, 20, 3, 10, 10, 6, 3, 21, 4, 10, 6, 10, 3, 15, 6, 15, 6, 6, 3, 50, 3, 6, 10, 8, 6, 20, 3, 10, 6, 20, 3, 35, 3, 6, 10, 10, 6, 20, 3, 21, 6, 6, 3, 50, 6, 6, 6, 15, 3, 50, 6

Offset: 1

Author

Yuval Dekel (dekelyuval(AT)hotmail.com), Aug 17 2004

Keywords

• nonn

Comments

The divisor lattice D(n) is the lattice of the divisors of the natural number n.

The empty set is counted as an antichain in D(n).

a(n) = gamma(n+1) where gamma is degree of cardinal completeness of Łukasiewicz n-valued logic. - Artur Jasinski, Mar 01 2010

References

• Alexander S. Karpenko, Lukasiewicz's Logics and Prime Numbers, Luniver Press, Beckington, 2006. See Table I p. 113.

Links

• Arkadiusz Wesolowski, Table of n, a(n) for n = 1..990
• Index entries for sequences related to Łukasiewicz

Crossrefs

Cf. A096825, A096826, A097699.

Cf. A175177, A175178. - Artur Jasinski, Mar 01 2010

Cf. A000005, A008480, A253249, A285572 A285573.

Programs

Mathematica

nn=200;
stableSets[u_,Q_]:=If[Length[u]===0,{{}},With[{w=First[u]},Join[stableSets[DeleteCases[u,w],Q],Prepend[#,w]&/@stableSets[DeleteCases[u,r_/;r===w||Q[r,w]||Q[w,r]],Q]]]];
Table[Length[stableSets[Divisors[n],Divisible]],{n,nn}] (* Gus Wiseman, Aug 24 2018 *)

Formula

a(n) = A285573(n) + 1. - Gus Wiseman, Aug 24 2018

Extensions

More terms from John W. Layman, Aug 20 2004