cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

A308957 Sum of the fourth largest parts in the partitions of n into 7 squarefree parts.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 2, 5, 7, 11, 14, 20, 24, 36, 41, 56, 64, 86, 98, 129, 147, 193, 222, 284, 324, 409, 457, 567, 635, 773, 862, 1037, 1147, 1375, 1516, 1778, 1953, 2290, 2510, 2920, 3186, 3680, 4017, 4614, 4996, 5734, 6226, 7081, 7682, 8732, 9450
Offset: 0

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Author

Wesley Ivan Hurt, Jul 03 2019

Keywords

Crossrefs

Programs

  • Mathematica
    Table[Sum[Sum[Sum[Sum[Sum[Sum[k * MoebiusMu[o]^2 * MoebiusMu[m]^2 * MoebiusMu[l]^2 * MoebiusMu[k]^2 * MoebiusMu[j]^2 * MoebiusMu[i]^2 * MoebiusMu[n - i - j - k - l - m - o]^2, {i, j, Floor[(n - j - k - l - m - o)/2]}], {j, k, Floor[(n - k - l - m - o)/3]}], {k, l, Floor[(n - l - m - o)/4]}], {l, m, Floor[(n - m - o)/5]}], {m, o, Floor[(n - o)/6]}], {o, Floor[n/7]}], {n, 0, 50}]
    Table[Total[Select[IntegerPartitions[n,{7}],AllTrue[#,SquareFreeQ]&][[;;,4]]],{n,0,60}] (* Harvey P. Dale, Sep 24 2023 *)

Formula

a(n) = Sum_{o=1..floor(n/7)} Sum_{m=o..floor((n-o)/6)} Sum_{l=m..floor((n-m-o)/5)} Sum_{k=l..floor((n-l-m-o)/4)} Sum_{j=k..floor((n-k-l-m-o)/3)} Sum_{i=j..floor((n-j-k-l-m-o)/2)} mu(o)^2 * mu(m)^2 * mu(l)^2 * mu(k)^2 * mu(j)^2 * mu(i)^2 * mu(n-i-j-k-l-m-o)^2 * k, where mu is the Möbius function (A008683).
a(n) = A308953(n) - A308954(n) - A308955(n) - A308956(n) - A308958(n) - A308959(n) - A308960(n).