%I #21 Feb 02 2021 15:28:12

%S 0,0,0,0,1,1,1,0,1,2,2,2,3,3,3,2,3,2,3,3,4,4,4,3,4,5,5,6,7,7,7,6,7,7,

%T 7,6,7,7,7,6,7,7,7,7,8,9,9,8,9,9,10,10,11,10,11,10,11,11,11,11,12,12,

%U 12,12,13,13,13,13,14,14,14,13,14,14,14,15,16

%N Number of partitions of n into two distinct parts such that the larger part is nonsquarefree.

%C Number of distinct rectangles with nonsquarefree length and integer width such that L + W = n, W < L. For example, a(8) = 0 since none of the rectangles 1 X 7, 2 X 6 and 3 X 5 (4 X 4 is not considered since we have W < L) have nonsquarefree length, i.e., 7,6,5 are all squarefree. a(10) = 2 since there are two rectangles with nonsquarefree length, namely 1 X 9 and 2 X 8.

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

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

%e a(14) = 3; the partitions of 14 into two parts are (13,1), (12,2), (11,3), (10,4), (9,5), (8,6), (7,7). Three of the larger parts of these partitions are nonsquarefree: 12, 9 and 8, so a(14) = 3.

%p A295284:=n->add(1-numtheory[mobius](n-i)^2, i=1..floor((n-1)/2)): seq(A295284(n), n=1..100);

%t Table[Sum[1 - MoebiusMu[n - k]^2, {k, Floor[(n - 1)/2]}], {n, 80}]

%t Table[Count[IntegerPartitions[n,{2}],_?(#[[1]]!=#[[2]]&&!SquareFreeQ[ #[[1]]]&)],{n,80}] (* _Harvey P. Dale_, Feb 02 2021 *)

%o (PARI) a(n) = sum(i=1, floor((n-1)/2), 1 - moebius(n-i)^2) \\ _Iain Fox_, Dec 08 2017

%Y Cf. A008683, A008966, A294235.

%K nonn,easy

%O 1,10

%A _Wesley Ivan Hurt_, Nov 19 2017