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A294060 Sum of the differences of the larger and smaller parts in the partitions of n into two parts with the smaller part squarefree. 2
0, 0, 1, 2, 4, 6, 9, 12, 15, 18, 22, 26, 31, 36, 42, 48, 54, 60, 66, 72, 79, 86, 94, 102, 110, 118, 127, 136, 146, 156, 167, 178, 189, 200, 212, 224, 236, 248, 261, 274, 287, 300, 314, 328, 343, 358, 374, 390, 406, 422, 438, 454, 471, 488, 505, 522, 539, 556 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,4

COMMENTS

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

LINKS

Muniru A Asiru, Table of n, a(n) for n = 1..1000

Index entries for sequences related to partitions

FORMULA

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

EXAMPLE

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

MAPLE

with(numtheory):

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

MATHEMATICA

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

PROG

(PARI) for(n=1, 50, print1(sum(k=1, floor(n/2), (n-2*k)*moebius(k)^2), ", ")) \\ G. C. Greubel, Mar 27 2018

CROSSREFS

Cf. A008683, A008966, A294061.

Sequence in context: A337134 A185601 A157795 * A003066 A075349 A156024

Adjacent sequences:  A294057 A294058 A294059 * A294061 A294062 A294063

KEYWORD

nonn,easy

AUTHOR

Wesley Ivan Hurt, Oct 22 2017

STATUS

approved

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Last modified July 29 04:31 EDT 2021. Contains 346340 sequences. (Running on oeis4.)