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A280226 Number of partitions of 2n into two squarefree parts.

%I A280226 #39 Dec 11 2023 21:52:18
%S A280226 1,2,2,3,2,4,3,5,4,6,5,7,5,7,5,8,7,11,7,11,8,13,8,13,8,14,10,13,11,15,
%T A280226 11,15,11,18,13,21,14,20,13,20,13,22,14,23,17,23,17,24,17,25,18,26,19,
%U A280226 31,19,29,20,31,20,31,20,33,23,30,23,32,23,32,23,35,24,41,25,39
%N A280226 Number of partitions of 2n into two squarefree parts.
%H A280226 Charles R Greathouse IV, <a href="/A280226/b280226.txt">Table of n, a(n) for n = 1..10000</a>
%H A280226 <a href="/index/Par#part">Index entries for sequences related to partitions</a>
%F A280226 a(n) = Sum_{i=1..n} mu(i)^2 * mu(2n-i)^2, where mu is the Möbius function (A008683).
%F A280226 a(n) = n - A302391(n). - _Wesley Ivan Hurt_, Dec 11 2023
%e A280226 From _Wesley Ivan Hurt_, Feb 20 2018: (Start)
%e A280226 a(5) = 2; there are two partitions of 2*5 = 10 into two squarefree parts: (7,3), (5,5).
%e A280226 a(6) = 4; there are four partitions of 2*6 = 12 into two squarefree parts: (11,1), (10,2), (7,5), (6,6).
%e A280226 a(7) = 3; there are three partitions of 2*7 = 14 into two squarefree parts: (13,1), (11,3), (7,7).
%e A280226 a(8) = 5; there are five partitions of 2*8 = 16 into two squarefree parts: (15,1), (14,2), (13,3), (11,5), (10,6). (End)
%p A280226 with(numtheory): A280226:=n->sum(mobius(i)^2*mobius(2*n-i)^2, i=1..n): seq(A280226(n), n=1..100);
%t A280226 f[n_] := Sum[(MoebiusMu[i]*MoebiusMu[2n -i])^2, {i, n}]; Array[f, 74] (* _Robert G. Wilson v_, Dec 29 2016 *)
%o A280226 (PARI) a(n)=sum(i=1,n, issquarefree(i) && issquarefree(2*n-i)) \\ _Charles R Greathouse IV_, Nov 05 2017
%Y A280226 Cf. A008683, A045917, A262991, A280250, A280251, A280252, A294248, A302391.
%K A280226 nonn,easy
%O A280226 1,2
%A A280226 _Wesley Ivan Hurt_, Dec 29 2016