A294061 Sum of the differences of the larger and smaller parts in the partitions of n into two parts with the larger part squarefree.

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

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 such that n-x is squarefree for x in 0 < x <= floor(n/2). For example, d/dx n*x-x^2 = n-2x. So for a(12), the integer values of x which make 12-x squarefree are x=1,2,5,6 and so a(12) = 12-2*1 + 12-2*2 + 12-2*5 + 12-2*6 = 10 + 8 + 2 + 0 = 20. - 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(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((n-2*i)*mobius(n-i)^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, (n-2*i)*issquarefree(n-i)); \\ Michel Marcus, Mar 24 2018

Crossrefs

Cf. A008683, A008966, A294060.

Sequence in context: A008301 A363386 A294104 * A113820 A319479 A303632

Adjacent sequences: A294058 A294059 A294060 * A294062 A294063 A294064

Keyword

nonn,easy

Author

Wesley Ivan Hurt, Oct 22 2017

Status

approved