A326626 -id:A326626 - cp's OEIS frontend

A326637 Sum of the largest parts of the partitions of n into 10 squarefree parts.

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 5, 5, 13, 19, 34, 40, 60, 82, 126, 153, 219, 275, 385, 464, 621, 738, 996, 1168, 1514, 1780, 2287, 2643, 3302, 3839, 4743, 5456, 6638, 7605, 9225, 10479, 12512, 14199, 16929, 19061, 22453, 25300, 29690, 33283, 38715, 43333

Offset: 0

Wesley Ivan Hurt, Jul 14 2019

nonn

Index entries for sequences related to partitions

Cf. A008683, A326626, A326627, A326628, A326629, A326630, A326631, A326632, A326633, A326634, A326635, A326636.

Table[Sum[Sum[Sum[Sum[Sum[Sum[Sum[Sum[Sum[(n-i-j-k-l-m-o-p-q-r) * MoebiusMu[r]^2 * MoebiusMu[q]^2 * MoebiusMu[p]^2 * 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 - p - q - r]^2 , {i, j, Floor[(n - j - k - l - m - o - p - q - r)/2]}], {j, k, Floor[(n - k - l - m - o - p - q - r)/3]}], {k, l, Floor[(n - l - m - o - p - q - r)/4]}], {l, m, Floor[(n - m - o - p - q - r)/5]}], {m, o, Floor[(n - o - p - q - r)/6]}], {o, p, Floor[(n - p - q - r)/7]}], {p, q, Floor[(n - q - r)/8]}], {q, r, Floor[(n - r)/9]}], {r, Floor[n/10]}], {n, 0, 50}]

a(n) = Sum_{r=1..floor(n/10)} Sum_{q=r..floor((n-r)/9)} Sum_{p=q..floor((n-q-r)/8)} Sum_{o=p..floor((n-p-q-r)/7)} Sum_{m=o..floor((n-o-p-q-r)/6)} Sum_{l=m..floor((n-m-o-p-q-r)/5)} Sum_{k=l..floor((n-l-m-o-p-q-r)/4)} Sum_{j=k..floor((n-k-l-m-o-p-q-r)/3)} Sum_{i=j..floor((n-j-k-l-m-o-p-q-r)/2)} mu(r)^2 * mu(q)^2 * mu(p)^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-p-q-r)^2 * (n-i-j-k-l-m-o-p-q-r), where mu is the Möbius function (A008683).

a(n) = A326627(n) - A326628(n) - A326629(n) - A326630(n) - A326631(n) - A326632(n) - A326633(n) - A326634(n) - A326635(n) - A326636(n).

A341070 Number of ways to write n as an ordered sum of 10 squarefree numbers.

Original entry on oeis.org

1, 10, 55, 210, 625, 1552, 3400, 6840, 12960, 23330, 40028, 65740, 104230, 160670, 241640, 354772, 509620, 718980, 999645, 1370720, 1853903, 2476250, 3274110, 4289810, 5568820, 7162184, 9138045, 11579180, 14574755, 18215900, 22619016, 27929990, 34311845, 41921710, 50946945

Offset: 10

Ilya Gutkovskiy, Feb 04 2021

nonn

Cf. A005117, A008683, A008966, A098235, A280210, A326626, A341064, A341065, A341066, A341067, A341068, A341069.

b:= proc(n, t) option remember;
      `if`(n=0, `if`(t=0, 1, 0), `if`(t<1, 0, add(
      `if`(numtheory[issqrfree](j), b(n-j, t-1), 0), j=1..n)))
    end:
a:= n-> b(n, 10):
seq(a(n), n=10..44);  # Alois P. Heinz, Feb 04 2021

nmax = 44; CoefficientList[Series[Sum[MoebiusMu[k]^2 x^k, {k, 1, nmax}]^10, {x, 0, nmax}], x] // Drop[#, 10] &

G.f.: (Sum_{k>=1} mu(k)^2 * x^k)^10.

A341098 Number of partitions of n into 10 distinct squarefree parts.

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 2, 4, 6, 8, 7, 10, 14, 17, 17, 22, 32, 35, 37, 47, 62, 71, 72, 91, 114, 132, 136, 167, 205, 234, 247, 293, 355, 398, 426, 497, 590, 661, 708, 819, 956, 1066, 1141, 1306, 1501, 1672, 1791, 2030, 2318, 2559, 2747, 3081, 3490, 3835, 4115

Offset: 72

Ilya Gutkovskiy, Feb 04 2021

nonn

Cf. A005117, A008966, A098236, A307835, A326626, A341070, A341073, A341074, A341075, A341095, A341096, A341097.

b:= proc(n, i, t) option remember; `if`(n=0,
      `if`(t=0, 1, 0), `if`(i<1 or t<1, 0, b(n, i-1, t)+
      `if`(numtheory[issqrfree](i), b(n-i, min(n-i, i-1), t-1), 0)))
    end:
a:= n-> b(n$2, 10):
seq(a(n), n=72..126);  # Alois P. Heinz, Feb 04 2021

b[n_, i_, t_] := b[n, i, t] = If[n == 0,
     If[t == 0, 1, 0], If[i < 1 || t < 1, 0, b[n, i - 1, t] +
     If[SquareFreeQ[i], b[n - i, Min[n - i, i - 1], t - 1], 0]]];
a[n_] := b[n, n, 10];
Table[a[n], {n, 72, 126}] (* Jean-François Alcover, Feb 24 2022, after Alois P. Heinz *)

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A326637 Sum of the largest parts of the partitions of n into 10 squarefree parts.

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A341070 Number of ways to write n as an ordered sum of 10 squarefree numbers.

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A341098 Number of partitions of n into 10 distinct squarefree parts.

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