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Sum of the larger parts of the partitions of n into two squarefree parts.
2

%I #35 Apr 26 2022 16:34:55

%S 0,1,2,5,3,8,11,18,13,12,16,34,28,31,37,63,50,56,44,88,59,83,73,129,

%T 93,91,100,138,105,103,123,195,151,173,169,303,201,199,219,345,255,

%U 256,298,442,341,274,289,482,380,294,255,525,401,410,270,539,422,487

%N Sum of the larger parts of the partitions of n into two squarefree parts.

%H Harvey P. Dale, <a href="/A261927/b261927.txt">Table of n, a(n) for n = 1..1000</a>

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

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

%F a(n) = A262351(n) - A261985(n).

%e a(4) = 5. There are two partitions of 4 into two squarefree parts: (3, 1) and (2, 2). The sum of the larger parts of these partitions is 3 + 2 = 5.

%e a(5) = 3. There is only one partition of 5 into two squarefree parts: (3, 2). The larger part is 3, thus a(5) = 3.

%p with(numtheory): A261987:=n->add((n-i)*mobius(i)^2*mobius(n-i)^2, i=1..floor(n/2)): seq(A261987(n), n=1..70);

%t Table[Sum[(n - i) MoebiusMu[i]^2 * MoebiusMu[n - i]^2, {i, Floor[n/2]}], {n, 70}]

%t Table[Total[Select[IntegerPartitions[n,{2}],AllTrue[#,SquareFreeQ]&][[All,1]]],{n,60}] (* _Harvey P. Dale_, Apr 26 2022 *)

%Y Cf. A008683, A071068, A261985, A262351.

%K nonn,easy

%O 1,3

%A _Wesley Ivan Hurt_, Oct 02 2015