A308841 Sum of the smallest parts in the partitions of n into 5 squarefree parts.

0, 0, 0, 0, 0, 1, 1, 2, 2, 4, 6, 8, 8, 12, 14, 20, 19, 26, 30, 39, 39, 52, 57, 71, 72, 95, 100, 123, 125, 155, 166, 198, 200, 242, 256, 304, 306, 366, 383, 445, 453, 533, 556, 642, 652, 762, 786, 898, 914, 1048, 1091, 1236, 1261, 1434, 1487, 1671, 1695, 1919

Offset

0,8

Links

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

Index entries for sequences related to partitions

Formula

a(n) = Sum_{l=1..floor(n/5)} Sum_{k=l..floor((n-l)/4)} Sum_{j=k..floor((n-k-l)/3)} Sum_{i=j..floor((n-j-k-l)/2)} mu(l)^2 * mu(k)^2 * mu(j)^2 * mu(i)^2 * mu(n-i-j-k-l)^2 * l, where mu is the Möbius function (A008683).

a(n) = A308839(n) - A308842(n) - A308843(n) - A308844(n) - A308845(n).

Example

The partitions of n into 5 parts for n = 10, 11, ..
                                                       1+1+1+1+10
                                                        1+1+1+2+9
                                                        1+1+1+3+8
                                                        1+1+1+4+7
                                                        1+1+1+5+6
                                            1+1+1+1+9   1+1+2+2+8
                                            1+1+1+2+8   1+1+2+3+7
                                            1+1+1+3+7   1+1+2+4+6
                                            1+1+1+4+6   1+1+2+5+5
                                            1+1+1+5+5   1+1+3+3+6
                                1+1+1+1+8   1+1+2+2+7   1+1+3+4+5
                                1+1+1+2+7   1+1+2+3+6   1+1+4+4+4
                                1+1+1+3+6   1+1+2+4+5   1+2+2+2+7
                    1+1+1+1+7   1+1+1+4+5   1+1+3+3+5   1+2+2+3+6
                    1+1+1+2+6   1+1+2+2+6   1+1+3+4+4   1+2+2+4+5
                    1+1+1+3+5   1+1+2+3+5   1+2+2+2+6   1+2+3+3+5
        1+1+1+1+6   1+1+1+4+4   1+1+2+4+4   1+2+2+3+5   1+2+3+4+4
        1+1+1+2+5   1+1+2+2+5   1+1+3+3+4   1+2+2+4+4   1+3+3+3+4
        1+1+1+3+4   1+1+2+3+4   1+2+2+2+5   1+2+3+3+4   2+2+2+2+6
        1+1+2+2+4   1+1+3+3+3   1+2+2+3+4   1+3+3+3+3   2+2+2+3+5
        1+1+2+3+3   1+2+2+2+4   1+2+3+3+3   2+2+2+2+5   2+2+2+4+4
        1+2+2+2+3   1+2+2+3+3   2+2+2+2+4   2+2+2+3+4   2+2+3+3+4
        2+2+2+2+2   2+2+2+2+3   2+2+2+3+3   2+2+3+3+3   2+3+3+3+3
--------------------------------------------------------------------------
  n  |     10          11          12          13          14        ...
--------------------------------------------------------------------------
a(n) |      6           8           8          12          14        ...
--------------------------------------------------------------------------
- Wesley Ivan Hurt, Sep 16 2019

Mathematica

Table[Sum[Sum[Sum[Sum[l * MoebiusMu[l]^2*MoebiusMu[k]^2*MoebiusMu[j]^2* MoebiusMu[i]^2*MoebiusMu[n - i - j - k - l]^2, {i, j, Floor[(n - j - k - l)/2]}], {j, k, Floor[(n - k - l)/3]}], {k, l, Floor[(n - l)/4]}], {l, Floor[n/5]}], {n, 0, 50}]

Crossrefs

Cf. A008683, A308839, A308840, A308842, A308843, A308844, A308845.

Sequence in context: A348781 A194695 A194693 * A176528 A108891 A147570

Adjacent sequences: A308838 A308839 A308840 * A308842 A308843 A308844

Keyword

nonn

Author

Wesley Ivan Hurt, Jun 28 2019

Status

approved