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A294061 Sum of the differences of the larger and smaller parts in the partitions of n into two parts with the larger part squarefree. 2
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 (list; graph; refs; listen; history; text; internal format)
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
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
Sequence in context: A008301 A363386 A294104 * A113820 A319479 A303632
KEYWORD
nonn,easy
AUTHOR
Wesley Ivan Hurt, Oct 22 2017
STATUS
approved

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Last modified March 1 14:57 EST 2024. Contains 370433 sequences. (Running on oeis4.)