OFFSET
1,3
COMMENTS
Sum of the lengths of the distinct rectangles with squarefree width and positive integer length such that L + W = n, W < L. For example, a(14) = 53; the rectangles are 1 X 13, 2 X 12, 3 X 11, 5 X 9, 6 X 8. The sum of the lengths is then 13 + 12 + 11 + 9 + 8 = 53. - Wesley Ivan Hurt, Nov 12 2017
FORMULA
a(n) = Sum_{i=1..floor((n-1)/2)} (n - i) * mu(i)^2, where mu is the Möbius function (A008683).
EXAMPLE
a(5) = 7; the partitions of 5 into two distinct parts are (4,1) and (3,2). The smaller parts are both squarefree, so the sum of the larger parts is 4+3 = 7.
a(10) = 24; the partitions of 10 into two distinct parts are (9,1), (8,2), (7,3) and (6,4). Of the smaller parts, only 1, 2, and 3 are squarefree, so we add the larger parts of those partitions to get 9+8+7 = 24.
MATHEMATICA
Table[Sum[(n - i) MoebiusMu[i]^2, {i, Floor[(n-1)/2]}], {n, 60}]
PROG
(PARI) a(n) = sum(i=1, (n-1)\2, (n-i)*moebius(i)^2); \\ Michel Marcus, Nov 08 2017
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Wesley Ivan Hurt, Oct 26 2017
STATUS
approved