A392164 a(n) is the size of the largest subset S of {1,...,N} such that every element of S+S is squarefree.

1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8

Offset

1,5

Links

Table of n, a(n) for n=1..98.

Thomas Bloom, Problem #1109, Erdős Problems.

Paul Erdős and András Sárközy, On divisibility properties of integers of the form a+a', Acta Math. Hungar. (1987), 117--122.

Sergei Konyagin, Problems of the set of squarefree numbers, Izv. Ross. Akad. Nauk Ser. Mat. (2004), 63--90.

Formula

a(n) >> log(n) (Erdős and Sárközy).

a(n) << n^(3/4)*log(n) (Erdős and Sárközy).

a(n) >> log(log(n))*(log(n))^2 (Konyagin).

a(n) << n^(11/15+o(1)) (Konyagin).

Mathematica

a[n_]:=Length[FindClique[RelationGraph[SquareFreeQ[#1+#2]&&#1!=#2&, Select[Range[n], SquareFreeQ[2*#]&]]][[1]]]

Prog

(Python)
from itertools import combinations
from networkx import empty_graph, find_cliques
from sympy import factorint
def A392164(n):
    def is_squarefree(n): return max(factorint(n).values(), default=1)<2
    v = [e for e in list(range(1, n+1)) if is_squarefree(2*e)]
    G = empty_graph(v)
    G.add_edges_from((a, b) for a, b in combinations(v, 2) if is_squarefree(a+b))
    return max(len(c) for c in find_cliques(G)) # Chai Wah Wu, Jan 07 2026

Crossrefs

Cf. A392165 (records).

Sequence in context: A135660 A163858 A236362 * A133502 A341422 A296078

Adjacent sequences: A392161 A392162 A392163 * A392165 A392166 A392167

Keyword

nonn

Author

Elijah Beregovsky, Jan 02 2026

Status

approved