A294062 Sum of the differences of the larger and smaller parts in the partitions of 2n into two parts with the smaller part squarefree.

0, 2, 6, 12, 18, 26, 36, 48, 60, 72, 86, 102, 118, 136, 156, 178, 200, 224, 248, 274, 300, 328, 358, 390, 422, 454, 488, 522, 556, 592, 630, 670, 710, 752, 796, 842, 888, 936, 986, 1038, 1090, 1144, 1200, 1258, 1316, 1374, 1434, 1496, 1558, 1620, 1682, 1746

Offset

1,2

Comments

Sum of the slopes of the tangent lines along the left side of the parabola b(x) = 2*n*x-x^2 at squarefree values of x for x in 0 < x <= n. For example, d/dx 2*n*x-x^2 = 2n-2x. So for a(6), the squarefree values of x are x=1,2,3,5,6 and so a(6) = 12-2*1 + 12-2*2 + 12-2*3 + 12-2*5 + 12-2*6 = 10 + 8 + 6 + 2 = 26. - Wesley Ivan Hurt, Mar 25 2018

Links

Table of n, a(n) for n=1..52.

Index entries for sequences related to partitions

Formula

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

Example

For n = 4, 8 can be partitioned into two parts with the smaller part squarefree in three ways: 7 + 1, 6 + 2, and 5 + 3, so a(4) = (7 - 1) + (6 - 2) + (5 - 3) = 12. - Michael B. Porter, Mar 27 2018

Mathematica

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

Prog

(PARI) a(n) = 2 * sum(i=1, n, (n-i)*issquarefree(i)); \\ Michel Marcus, Mar 26 2018

Crossrefs

Cf. A008683, A008966.

Sequence in context: A330320 A267461 A108585 * A193764 A278484 A159793

Adjacent sequences: A294059 A294060 A294061 * A294063 A294064 A294065

Keyword

nonn,easy

Author

Wesley Ivan Hurt, Oct 22 2017

Status

approved