A309455 Number of squarefree parts in the partitions of n into 3 parts.

0, 0, 0, 3, 3, 6, 8, 11, 13, 18, 19, 24, 27, 33, 36, 44, 47, 54, 59, 66, 70, 81, 84, 95, 100, 111, 116, 128, 134, 146, 153, 165, 172, 186, 192, 207, 215, 230, 238, 256, 264, 281, 291, 309, 319, 340, 349, 369, 380, 400, 411, 432, 442, 464, 475, 497, 508, 532

Offset

0,4

Links

Table of n, a(n) for n=0..57.

Index entries for sequences related to partitions

Formula

a(n) = Sum_{j=1..floor(n/3)} Sum_{i=j..floor((n-j)/2)} (mu(i)^2 + mu(j)^2 + 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      3      6      8     11     13     18     19      ...
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Mathematica

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

Crossrefs

Cf. A008683.

Sequence in context: A021752 A049626 A241343 * A168637 A372887 A241390

Adjacent sequences: A309452 A309453 A309454 * A309456 A309457 A309458

Keyword

nonn

Author

Wesley Ivan Hurt, Aug 03 2019

Status

approved