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

Author

Wesley Ivan Hurt, Oct 22 2017

Keywords

• nonn
• easy

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

Examples

			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
		

Links

• Index entries for sequences related to partitions

Crossrefs

Cf. A008683 (mu), A008966, A013928, A066779.

Programs

Mathematica

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

PARI

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

Formula

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

a(n) = 2*(n*A013928(n) - A066779(n)). - Wesley Ivan Hurt, Jul 08 2025