A294060 Sum of the differences of the larger and smaller parts in the partitions of n into two parts with the smaller part squarefree.

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

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