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A373119 Cardinality of the largest subset of {1,...,n} such that no four distinct elements of this subset multiply to a square.

%I A373119 #47 Dec 30 2024 02:12:19
%S A373119 1,2,3,4,5,5,6,6,7,7,8,8,9,9,10,10,11,11,12,12,12,12,13,13,13,13,13,
%T A373119 13,14,14,15,15,15,15,16,16,17,17,17,17,18,18,19,19,19,19,20,20,20,20,
%U A373119 20,20,21,21,22,22,22,22,23,23,24,24,24,24,25,25,26,26,26
%N A373119 Cardinality of the largest subset of {1,...,n} such that no four distinct elements of this subset multiply to a square.
%C A373119 a(n) >= A000720(n).
%C A373119 a(n) ~ n/log n (Erdős-Sárközy-Sós). Best bounds currently are due to Pach-Vizer.
%C A373119 a(n+1)-a(n) is either 0 or 1 for any n. (Is equal to 1 when n+1 is prime.)
%C A373119 If "four" is replaced by "one", "two", "three", "five", or "any odd", one obtains A028391, A013928, A372306, A373178, and A373114 respectively.
%H A373119 P. Erdős, A. Sárközy, and V. T. Sós, <a href="https://doi.org/10.1016/0195-6698(95)90039-X">On Product Representations of Powers, I</a>, Europ. J. Combinatorics 16 (1995), 567-588.
%H A373119 P. Pach and M. Vizer, <a href="https://doi.org/10.37236/11477">Improved Lower Bounds for Multiplicative Square-Free Sequences</a>, The Electronic Journal of Combinatorics, Volume 30, Issue 4 (2023), P4.31.
%H A373119 Terence Tao, <a href="https://arxiv.org/abs/2405.11610">On product representations of squares</a>, arXiv:2405.11610 [math.NT], May 2024.
%e A373119 a(7)=6, because the set {1,2,3,4,5,7} has no four distinct elements multiplying to a square, but {1,2,3,4,5,6,7} has 1*2*3*6 = 6^2.
%o A373119 (Python)
%o A373119 from math import isqrt
%o A373119 def is_square(n):
%o A373119     return isqrt(n) ** 2 == n
%o A373119 def valid_subset(A):
%o A373119     length = len(A)
%o A373119     for i in range(length):
%o A373119         for j in range(i + 1, length):
%o A373119             for k in range(j + 1, length):
%o A373119                 for l in range(k + 1, length):
%o A373119                     if is_square(A[i] * A[j] * A[k] * A[l]):
%o A373119                         return False
%o A373119     return True
%o A373119 def largest_subset_size(N):
%o A373119     from itertools import combinations
%o A373119     for size in reversed(range(1, N + 1)):
%o A373119         for subset in combinations(range(1, N + 1), size):
%o A373119             if valid_subset(subset):
%o A373119                 return size
%o A373119 for N in range(1, 23):
%o A373119     print(largest_subset_size(N))
%o A373119 (Python)
%o A373119 from math import prod
%o A373119 from functools import lru_cache
%o A373119 from itertools import combinations
%o A373119 from sympy.ntheory.primetest import is_square
%o A373119 @lru_cache(maxsize=None)
%o A373119 def A373119(n):
%o A373119     if n==1: return 1
%o A373119     i = A373119(n-1)+1
%o A373119     if sum(1 for p in combinations(range(1,n),3) if is_square(n*prod(p))) > 0:
%o A373119         a = [set(p) for p in combinations(range(1,n+1),4) if is_square(prod(p))]
%o A373119         for q in combinations(range(1,n),i-1):
%o A373119             t = set(q)|{n}
%o A373119             if not any(s<=t for s in a):
%o A373119                 return i
%o A373119         else:
%o A373119             return i-1
%o A373119     else:
%o A373119         return i # _Chai Wah Wu_, May 30 2024
%Y A373119 Cf. A028391, A013928, A372306, A373114, A373178, A373195.
%Y A373119 Lower bounded by A000720.
%K A373119 nonn
%O A373119 1,2
%A A373119 _Terence Tao_, May 26 2024
%E A373119 a(22)-a(37) from _Michael S. Branicky_, May 26 2024
%E A373119 a(38)-a(63) from _Martin Ehrenstein_, May 27 2024
%E A373119 a(64)-a(69) from _Jinyuan Wang_, Dec 30 2024