A309456 Number of squarefree parts in the partitions of n into 4 parts.

0, 0, 0, 0, 4, 4, 8, 11, 19, 22, 32, 38, 51, 59, 75, 86, 108, 123, 147, 167, 197, 218, 254, 281, 322, 354, 400, 437, 491, 534, 592, 643, 710, 765, 840, 903, 984, 1055, 1145, 1222, 1324, 1410, 1517, 1614, 1734, 1837, 1968, 2083, 2222, 2348, 2499, 2633, 2797

Offset

0,5

Links

Table of n, a(n) for n=0..52.

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)} (mu(i)^2 + mu(j)^2 + mu(k)^2 + 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) |     19          22          32          38          51        ...
--------------------------------------------------------------------------
- Wesley Ivan Hurt, Sep 07 2019

Mathematica

Table[Sum[Sum[Sum[(MoebiusMu[i]^2 + MoebiusMu[j]^2 + MoebiusMu[k]^2 + 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}]
Table[Count[Flatten[IntegerPartitions[n, {4}]], _?SquareFreeQ], {n, 0, 60}] (* Harvey P. Dale, Apr 17 2021 *)

Crossrefs

Cf. A008683.

Sequence in context: A152967 A376433 A301705 * A004024 A292276 A278083

Adjacent sequences: A309453 A309454 A309455 * A309457 A309458 A309459

Keyword

nonn

Author

Wesley Ivan Hurt, Aug 03 2019

Status

approved