A294145 -id:A294145 - cp's OEIS frontend

A294146 Sum of the larger parts of the partitions of n into two parts with smaller part squarefree.

0, 1, 2, 5, 7, 12, 15, 18, 21, 29, 33, 43, 48, 60, 66, 72, 78, 84, 90, 106, 113, 131, 139, 147, 155, 176, 185, 208, 218, 243, 254, 265, 276, 304, 316, 328, 340, 371, 384, 397, 410, 444, 458, 494, 509, 547, 563, 579, 595, 611, 627, 669, 686, 703, 720, 737

Offset: 1

Wesley Ivan Hurt, Oct 23 2017

Harvey P. Dale, Table of n, a(n) for n = 1..1000
Index entries for sequences related to partitions

Cf. A008683, A008966, A294145.

Table[Sum[(n - i)*MoebiusMu[i]^2, {i, Floor[n/2]}], {n, 80}]
Table[Total[Select[IntegerPartitions[n,{2}],SquareFreeQ[#[[2]]]&][[;;,1]]],{n,70}] (* Harvey P. Dale, Mar 23 2023 *)

a(n) = Sum_{i=1..floor(n/2)} (n - i) * mu(i)^2, where mu is the Möbius function (A008683).

A294284 Sum of the smaller parts of the partitions of n into two distinct parts with larger part squarefree.

Original entry on oeis.org

0, 0, 1, 1, 2, 1, 3, 6, 9, 7, 10, 8, 11, 8, 12, 17, 22, 28, 34, 31, 37, 33, 40, 48, 56, 51, 59, 53, 60, 53, 61, 70, 79, 72, 82, 93, 104, 97, 109, 122, 135, 128, 142, 135, 149, 140, 154, 169, 184, 199, 214, 204, 219, 235, 251, 268, 285, 274, 292, 281

Offset: 1

Wesley Ivan Hurt, Oct 26 2017

Sum of the widths of the distinct rectangles with squarefree length and positive integer width such that L + W = n, W < L. For example, a(13) = 11; the rectangles are 2 X 11, 3 X 10, 6 X 7. The sum of the widths is then 2 + 3 + 6 = 11. - Wesley Ivan Hurt, Nov 12 2017

Index entries for sequences related to partitions

Cf. A008683, A294145, A294285.

Table[Sum[i*MoebiusMu[n - i]^2, {i, Floor[(n-1)/2]}], {n, 60}]

a(n) = sum(i=1, (n-1)\2, i*moebius(n-i)^2); \\ Michel Marcus, Nov 05 2017

a(n) = Sum_{i=1..floor((n-1)/2)} i * mu(n-i)^2, where mu is the Möbius function (A008683).

cp's OEIS Frontend

A294146 Sum of the larger parts of the partitions of n into two parts with smaller part squarefree.

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