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A294062
Sum of the differences of the larger and smaller parts in the partitions of 2n into two parts with the smaller part squarefree.
1
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
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
Sequence in context: A330320 A267461 A108585 * A193764 A278484 A159793
KEYWORD
nonn,easy
AUTHOR
Wesley Ivan Hurt, Oct 22 2017
STATUS
approved