A358370 a(n) is the size of the largest 3-independent set in the cyclic group Zn.

0, 0, 0, 1, 1, 1, 1, 2, 1, 2, 2, 3, 2, 3, 3, 4, 3, 4, 3, 5, 3, 5, 4, 6, 5, 6, 4, 7, 5, 7, 5, 8, 6, 8, 7, 9, 6, 9, 6, 10, 7, 10, 7, 11, 9, 11, 8, 12, 8, 12, 9, 13, 9, 13, 11, 14, 9, 14, 10, 15, 10, 15, 10, 16, 13, 16, 11, 17, 12, 17, 12, 18, 12, 18, 15, 19, 14

Offset: 1

Author

Stefano Spezia, Nov 12 2022

Keywords

• nonn
• easy

Links

• Béla Bajnok, Additive Combinatorics in Groups and Geometric Combinatorics on Spheres, arXiv:2211.01890 [math.NT], 2022. See p. 4.
• Béla Bajnok and Imre Ruzsa, The independence number of a subset of an abelian group. Integers, 3 (2003), Paper A2. See p. 5.

Crossrefs

Cf. A002265, A007528, A027750, A152467.

Programs

Mathematica

b[n_]:=Min[Intersection[Divisors[n], Select[Prime[Range[PrimePi[n]]], Mod[#, 6]==5 &]]]; a[n_]:=If[EvenQ[n], Floor[n/4], If[IntegerQ[b[n]], (1+1/b[n])n/6, Floor[n/6]]]; Array[a,80]

Formula

a(n) = floor(n/4) if n is even, a(n) = (1 + 1/p)*n/6 if n is odd with smallest prime divisor p congruent 5 mod 6, and a(n) = floor(n/6) otherwise.