A309479 Sum of the squarefree parts of the partitions of n into 4 parts.

0, 0, 0, 0, 4, 5, 12, 17, 36, 46, 74, 93, 135, 169, 232, 278, 384, 462, 587, 702, 881, 1009, 1250, 1432, 1728, 1976, 2338, 2637, 3091, 3481, 3987, 4466, 5114, 5658, 6420, 7091, 7967, 8775, 9813, 10729, 11983, 13089, 14467, 15771, 17422, 18890, 20784, 22504

Offset

0,5

Links

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

Index entries for sequences related to partitions

Formula

a(n) = Sum_{k=1..floor(n/4)} Sum_{j=k..floor((n-k)/3)} Sum_{i=j..floor((n-j-k)/2)} (i * mu(i)^2 + j * mu(j)^2 + k * mu(k)^2 + (n-i-j-k) * mu(n-i-j-k)^2), where mu is the Möbius function (A008683).

Example

Figure 1: The partitions of n into 4 parts for n = 8, 9, ..
                                                         1+1+1+9
                                                         1+1+2+8
                                                         1+1+3+7
                                                         1+1+4+6
                                             1+1+1+8     1+1+5+5
                                             1+1+2+7     1+2+2+7
                                 1+1+1+7     1+1+3+6     1+2+3+6
                                 1+1+2+6     1+1+4+5     1+2+4+5
                                 1+1+3+5     1+2+2+6     1+3+3+5
                     1+1+1+6     1+1+4+4     1+2+3+5     1+3+4+4
         1+1+1+5     1+1+2+5     1+2+2+5     1+2+4+4     2+2+2+6
         1+1+2+4     1+1+3+4     1+2+3+4     1+3+3+4     2+2+3+5
         1+1+3+3     1+2+2+4     1+3+3+3     2+2+2+5     2+2+4+4
         1+2+2+3     1+2+3+3     2+2+2+4     2+2+3+4     2+3+3+4
         2+2+2+2     2+2+2+3     2+2+3+3     2+3+3+3     3+3+3+3
--------------------------------------------------------------------------
  n  |      8           9          10          11          12        ...
--------------------------------------------------------------------------
a(n) |     36          46          74          93         135        ...
--------------------------------------------------------------------------
- Wesley Ivan Hurt, Sep 07 2019

Mathematica

Table[Sum[Sum[Sum[(i*MoebiusMu[i]^2 + j*MoebiusMu[j]^2 + k*MoebiusMu[k]^2 + (n - i - j - k)*MoebiusMu[n - i - j - k]^2), {i, j, Floor[(n - j - k)/2]}], {j, k, Floor[(n - k)/3]}], {k, Floor[n/4]}], {n, 0, 50}]

Crossrefs

Cf. A008683, A309456, A309478, A309479, A309480, A309481, A309482, A309484, A309485, A309486.

Sequence in context: A261692 A131328 A054451 * A352720 A369697 A308775

Adjacent sequences: A309476 A309477 A309478 * A309480 A309481 A309482

Keyword

nonn

Author

Wesley Ivan Hurt, Aug 04 2019

Status

approved