A309478 Sum of the squarefree parts of the partitions of n into 3 parts.

0, 0, 0, 3, 4, 10, 14, 24, 32, 51, 56, 77, 95, 128, 146, 198, 225, 280, 318, 384, 416, 524, 549, 672, 726, 861, 915, 1063, 1143, 1292, 1382, 1551, 1661, 1867, 1966, 2211, 2355, 2618, 2762, 3094, 3263, 3602, 3798, 4185, 4409, 4869, 5078, 5524, 5794, 6264

Offset

0,4

Links

David A. Corneth, Table of n, a(n) for n = 0..9999

Index entries for sequences related to partitions

Formula

a(n) = Sum_{j=1..floor(n/3)} Sum_{i=j..floor((n-j)/2)} (i * mu(i)^2 + j * mu(j)^2 + (n-i-j) * mu(n-i-j)^2), where mu is the Möbius function (A008683).

Example

Figure 1: The partitions of n into 3 parts for n = 3, 4, ...
                                                          1+1+8
                                                   1+1+7  1+2+7
                                                   1+2+6  1+3+6
                                            1+1+6  1+3+5  1+4+5
                                     1+1+5  1+2+5  1+4+4  2+2+6
                              1+1+4  1+2+4  1+3+4  2+2+5  2+3+5
                       1+1+3  1+2+3  1+3+3  2+2+4  2+3+4  2+4+4
         1+1+1  1+1+2  1+2+2  2+2+2  2+2+3  2+3+3  3+3+3  3+3+4    ...
-----------------------------------------------------------------------
  n  |     3      4      5      6      7      8      9     10      ...
-----------------------------------------------------------------------
a(n) |     3      4     10     14     24     32     51     56      ...
-----------------------------------------------------------------------

Mathematica

Table[Sum[Sum[i*MoebiusMu[i]^2 + j*MoebiusMu[j]^2 + (n - i - j) MoebiusMu[n - i - j]^2, {i, j, Floor[(n - j)/2]}], {j, Floor[n/3]}], {n, 0, 50}]
Table[Total[Select[Flatten[IntegerPartitions[n, {3}]], SquareFreeQ]], {n, 0, 50}] (* Harvey P. Dale, Apr 09 2022 *)

Crossrefs

Cf. A008683, A309455, A309478, A309479, A309480, A309481, A309482, A309484, A309485, A309486.

Sequence in context: A035594 A167273 A096380 * A329805 A071019 A173285

Adjacent sequences: A309475 A309476 A309477 * A309479 A309480 A309481

Keyword

nonn

Author

Wesley Ivan Hurt, Aug 04 2019

Status

approved