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A294283 Sum of the larger parts of the partitions of n into two distinct parts with smaller part squarefree. 0
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified December 10 15:09 EST 2019. Contains 329896 sequences. (Running on oeis4.)