

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((n1)/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[(n1)/2]}], {n, 60}]


PROG

(PARI) a(n) = sum(i=1, (n1)\2, (ni)*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



