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A294098 Number of partitions of 2n into two parts such that one part is squarefree and the other part is nonsquarefree. 1

%I

%S 0,0,1,0,3,1,4,1,4,2,5,2,7,4,9,4,8,1,11,4,12,4,14,5,15,5,13,8,14,8,17,

%T 9,19,7,18,3,19,8,23,10,25,9,26,9,22,12,25,12,27,11,27,12,28,5,31,12,

%U 32,12,34,13,36,12,31,18,34,18,37,19,39,17,40,7,41

%N Number of partitions of 2n into two parts such that one part is squarefree and the other part is nonsquarefree.

%H <a href="/index/Par#part">Index entries for sequences related to partitions</a>

%F a(n) = n - Sum_{i=1..n} [c(i) = c(2*n-i)], where [] is the Iverson bracket and c is the squarefree characteristic (A008966).

%F a(n) = Sum_{i=1..n} mu(i)^2 * (1-mu(2*n-i)^2) + (1-mu(i)^2) * mu(2*n-i)^2, where mu is the Möbius function (A008683). - _Wesley Ivan Hurt_, Nov 18 2017

%t Table[n - Sum[KroneckerDelta[MoebiusMu[k]^2, MoebiusMu[2 n - k]^2], {k, n}], {n, 80}]

%t Table[Count[IntegerPartitions[2n,{2}],_?(Total[Boole[ SquareFreeQ/@#]] == 1&)],{n,80}] (* _Harvey P. Dale_, Jul 27 2021 *)

%Y Cf. A008683, A008966, A294097.

%K nonn,easy

%O 1,5

%A _Wesley Ivan Hurt_, Oct 22 2017

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Last modified September 20 05:24 EDT 2021. Contains 347577 sequences. (Running on oeis4.)