A294283 Sum of the larger parts of the partitions of n into two distinct parts with smaller part squarefree.

0, 0, 2, 3, 7, 9, 15, 18, 21, 24, 33, 37, 48, 53, 66, 72, 78, 84, 90, 96, 113, 120, 139, 147, 155, 163, 185, 194, 218, 228, 254, 265, 276, 287, 316, 328, 340, 352, 384, 397, 410, 423, 458, 472, 509, 524, 563, 579, 595, 611, 627, 643, 686, 703, 720, 737, 754

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

Links

Table of n, a(n) for n=1..57.

Index entries for sequences related to partitions

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

Cf. A008683, A008966, A262869, A262871, A294146.

Sequence in context: A014837 A019312 A135369 * A294122 A211539 A109660

Adjacent sequences: A294280 A294281 A294282 * A294284 A294285 A294286

Keyword

nonn,easy

Author

Wesley Ivan Hurt, Oct 26 2017

Status

approved