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Number of squarefree numbers among the parts of the partitions of n into two parts.
7

%I #22 Aug 18 2021 19:38:05

%S 0,2,2,4,3,5,5,6,6,7,7,9,8,10,10,11,11,12,12,14,13,15,15,16,16,17,17,

%T 18,17,19,19,20,20,22,22,23,23,25,25,26,26,28,28,30,29,30,30,31,31,31,

%U 31,33,32,33,33,34,34,36,36,38,37,39,39,39,39,41,41,43

%N Number of squarefree numbers among the parts of the partitions of n into two parts.

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

%F a(n) = Sum_{i=1..floor(n/2)} mu(i)^2 + mu(n-i)^2, where mu is the Möebius function (A008683).

%F a(n) = A262868(n) + A262869(n).

%e a(5)=3; there are 2 partitions of 5 into two parts: (4,1) and (3,2). Three of the parts in the partitions are squarefree, so a(5)=3.

%e a(6)=5; there are 3 partitions of 6 into two parts: (5,1), (4,2) and (3,3). Five of the parts in the partitions are squarefree, so a(6)=5.

%p with(numtheory): A262991:=n->add(mobius(i)^2+mobius(n-i)^2, i=1..floor(n/2)): seq(A262991(n), n=1..100);

%t Table[Sum[MoebiusMu[i]^2 + MoebiusMu[n - i]^2, {i, Floor[n/2]}], {n, 100}]

%t Table[Count[Flatten[IntegerPartitions[n,{2}]],_?SquareFreeQ],{n,70}] (* _Harvey P. Dale_, Aug 18 2021 *)

%o (PARI) vector(100, n, sum(k=1, n\2, moebius(k)^2 + moebius(n-k)^2)) \\ _Altug Alkan_, Oct 07 2015

%Y Cf. A008683, A071068, A261985, A262351, A262869, A262870, A262871, A262992.

%K nonn,easy

%O 1,2

%A _Wesley Ivan Hurt_, Oct 06 2015