

A294061


Sum of the differences of the larger and smaller parts in the partitions of n into two parts with the larger part squarefree.


2



0, 0, 1, 2, 1, 4, 8, 12, 9, 6, 13, 20, 17, 26, 36, 46, 41, 52, 46, 58, 52, 66, 81, 96, 88, 80, 98, 90, 83, 104, 126, 148, 139, 162, 186, 210, 199, 224, 250, 276, 263, 290, 318, 346, 332, 318, 350, 382, 367, 352, 337, 372, 357, 394, 378, 416, 399, 438, 478
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OFFSET

1,4


COMMENTS

Sum of the slopes of the tangent lines along the left side of the parabola b(x) = n*xx^2 such that nx is squarefree for x in 0 < x <= floor(n/2). For example, d/dx n*xx^2 = n2x. So for a(12), the integer values of x which make 12x squarefree are x=1,2,5,6 and so a(12) = 122*1 + 122*2 + 122*5 + 122*6 = 10 + 8 + 2 + 0 = 20.  Wesley Ivan Hurt, Mar 24 2018


LINKS



FORMULA

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


EXAMPLE

For n = 10, there are two partitions into a squarefree number and a smaller number, 7 + 3 and 6 + 4. So a(10) = (7  3) + (6  4) = 6.  Michael B. Porter, Apr 05 2018


MAPLE

with(numtheory):
seq(add((n2*i)*mobius(ni)^2, i=1..floor(n/2)), n=1..60); # Muniru A Asiru, Mar 24 2018


MATHEMATICA

Table[Sum[(n  2 i) MoebiusMu[n  i]^2, {i, Floor[n/2]}], {n, 80}]


PROG

(PARI) a(n) = sum(i=1, n\2, (n2*i)*issquarefree(ni)); \\ Michel Marcus, Mar 24 2018


CROSSREFS



KEYWORD

nonn,easy


AUTHOR



STATUS

approved



