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%I #28 Jun 19 2019 21:10:44
%S 0,0,1,2,4,6,9,12,15,18,22,26,31,36,42,48,54,60,66,72,79,86,94,102,
%T 110,118,127,136,146,156,167,178,189,200,212,224,236,248,261,274,287,
%U 300,314,328,343,358,374,390,406,422,438,454,471,488,505,522,539,556
%N Sum of the differences of the larger and smaller parts in the partitions of n into two parts with the smaller part squarefree.
%C Sum of the slopes of the tangent lines along the left side of the parabola b(x) = n*x-x^2 at squarefree values of x for x in 0 < x <= floor(n/2). For example, d/dx n*x-x^2 = n-2x. So for a(11), x=1,2,3,5 and so 11-2*1 + 11-2*2 + 11-2*3 + 11-2*5 = 9 + 7 + 5 + 1 = 22. - _Wesley Ivan Hurt_, Mar 24 2018
%H Muniru A Asiru, <a href="/A294060/b294060.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 - 2i) * mu(i)^2, where mu is the Möbius function (A008683).
%e For n = 9, there are three partitions of 9 into a number and a smaller squarefree number, 8 + 1, 7 + 2, and 6 + 3. So a(9) = (8 - 1) + (7 - 2) + (6 - 3) = 15. - _Michael B. Porter_, Mar 29 2018
%p with(numtheory):
%p seq(add((n-2*i)*mobius(i)^2,i=1..floor(n/2)),n=1..60); # _Muniru A Asiru_, Mar 24 2018
%t Table[Sum[(n - 2 i) MoebiusMu[i]^2, {i, Floor[n/2]}], {n, 80}]
%o (PARI) for(n=1,50, print1(sum(k=1,floor(n/2), (n-2*k)*moebius(k)^2), ", ")) \\ _G. C. Greubel_, Mar 27 2018
%Y Cf. A008683, A008966, A294061.
%K nonn,easy
%O 1,4
%A _Wesley Ivan Hurt_, Oct 22 2017
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