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%I #23 Apr 09 2023 14:10:00
%S 0,0,0,4,6,8,10,20,33,39,45,63,71,79,87,111,121,149,161,193,207,221,
%T 235,273,314,332,377,425,447,469,491,545,569,593,617,677,703,729,755,
%U 821,849,877,905,977,1052,1084,1116,1196,1279,1365,1403,1493,1533,1627
%N Sum of the larger parts of the partitions of 2n into two parts with smaller part nonsquarefree.
%H Robert Israel, <a href="/A294243/b294243.txt">Table of n, a(n) for n = 1..10000</a>
%H <a href="/index/Par#part">Index entries for sequences related to partitions</a>
%F a(n) = Sum_{i=1..n} (2*n - i) * (1 - mu(i)^2), where mu is the Möbius function (A008683).
%p N:= 100: # to get a(1)..a(N)
%p S:= ListTools:-PartialSums(map(t -> `if`(numtheory:-issqrfree(t),[0,0],[1,t]), [$1..N])):
%p seq(2*n*S[n,1]-S[n,2], n=1..N); # _Robert Israel_, Oct 27 2017
%t Table[Sum[(2 n - k) (1 - MoebiusMu[k]^2), {k, n}], {n, 80}]
%t Table[Total[Select[IntegerPartitions[2 n,{2}],!SquareFreeQ[#[[2]]]&][[;;,1]]],{n,60}] (* _Harvey P. Dale_, Apr 09 2023 *)
%o (PARI) a(n) = sum(i=1, n, (2*n-i)*(1-moebius(i)^2)); \\ _Michel Marcus_, Oct 27 2017
%Y Cf. A008683, A008966, A294244.
%K nonn,easy
%O 1,4
%A _Wesley Ivan Hurt_, Oct 25 2017
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