This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A345128 #21 Dec 20 2024 22:30:04
%S A345128 0,1,1,1,1,1,2,2,1,1,2,2,3,2,2,3,4,3,3,3,2,4,4,4,4,4,4,3,5,4,5,5,4,6,
%T A345128 4,5,7,6,5,6,8,5,9,7,6,7,8,7,8,7,5,8,10,7,6,8,7,10,9,7,11,10,8,10,8,8,
%U A345128 11,10,9,10,11,10,13,13,9,12,10,10,14,12,12,12,13,11,12
%N A345128 Number of squarefree products s*t from all positive integer pairs (s,t), such that s + t = n, s <= t.
%C A345128 From _Lei Zhou_, Dec 19 2024: (Start)
%C A345128 a(n) is also the total number of appearance of n in A379049, by definition.
%C A345128 Conjecture: a(n) > 0 for all n > 1. (End)
%C A345128 A simple case of Zhou's conjecture: a(p) > 0 where p is prime. With some work this can be extended to a(n) > 0 for n with sum_{p | n} 1/p < 1/10 or so (the limit of the method is 6/Pi^2 - 1/2, so it can't prove the full conjecture). See my comment in A071068. - _Charles R Greathouse IV_, Dec 20 2024
%F A345128 a(n) = Sum_{k=1..floor(n/2)} mu(k*(n-k))^2, where mu is the Möbius function (A008683).
%F A345128 a(n) <= A071068(n) and hence a(n) < 0.303967n for n > 3. - _Charles R Greathouse IV_, Dec 20 2024
%e A345128 a(13) = 3; The partitions of 13 into two positive integer parts (s,t) where s <= t are (1,12), (2,11), (3,10), (4,9), (5,8), (6,7). The corresponding products are 1*12, 2*11, 3*10, 4*9, 5*8, and 6*7; 3 of which are squarefree.
%t A345128 Table[Sum[MoebiusMu[k (n - k)]^2, {k, Floor[n/2]}], {n, 100}]
%o A345128 (PARI) a(n)=my(s); forsquarefree(k=1,n\2, gcd(n,k[1])==1 && issquarefree(n-k[1]) && s++); s \\ _Charles R Greathouse IV_, Dec 20 2024
%Y A345128 Cf. A008683 (mu), A379049, A071068.
%K A345128 nonn
%O A345128 1,7
%A A345128 _Wesley Ivan Hurt_, Jun 08 2021