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.

Showing 1-10 of 11 results. Next

A326626 Number of partitions of n into 10 squarefree parts.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 2, 4, 5, 8, 9, 13, 16, 23, 26, 35, 41, 54, 62, 79, 90, 115, 130, 161, 182, 224, 251, 303, 341, 408, 456, 539, 601, 709, 786, 915, 1014, 1179, 1299, 1496, 1649, 1892, 2078, 2368, 2597, 2953, 3230, 3645, 3986, 4492, 4895
Offset: 0

Views

Author

Wesley Ivan Hurt, Jul 14 2019

Keywords

Links

Crossrefs

Cf. A008683, A326627, A326628, A326629, A326630, A326631, A326632, A326633, A326634, A326635, A326636, A326637.

Programs

Formula

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, where mu is the Möbius function (A008683).
a(n) = A326627(n)/n for n > 0.

A326627 Sum of all the parts in the partitions of n into 10 squarefree parts.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 10, 11, 24, 26, 56, 75, 128, 153, 234, 304, 460, 546, 770, 943, 1296, 1550, 2054, 2430, 3220, 3770, 4830, 5642, 7168, 8283, 10302, 11935, 14688, 16872, 20482, 23439, 28360, 32226, 38430, 43602, 51876, 58455, 68816, 77503, 90816
Offset: 0

Views

Author

Wesley Ivan Hurt, Jul 14 2019

Keywords

Links

Crossrefs

Cf. A008683, A326626, A326628, A326629, A326630, A326631, A326632, A326633, A326634, A326635, A326636, A326637.

Programs

  • Mathematica
    Table[n * Sum[Sum[Sum[Sum[Sum[Sum[Sum[Sum[Sum[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}]

Formula

a(n) = 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, where mu is the Möbius function (A008683).
a(n) = n * A326626(n).
a(n) = A326628(n) + A326629(n) + A326630(n) + A326631(n) + A326632(n) + A326633(n) + A326634(n) + A326635(n) + A326636(n) + A326637(n).

A326628 Sum of the smallest parts of the partitions of n into 10 squarefree parts.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 2, 4, 5, 8, 9, 13, 16, 24, 27, 36, 43, 57, 66, 84, 96, 124, 142, 176, 199, 248, 280, 338, 382, 461, 519, 615, 688, 819, 914, 1066, 1186, 1392, 1539, 1780, 1970, 2278, 2512, 2879, 3167, 3629, 3986, 4517, 4960, 5631, 6159
Offset: 0

Views

Author

Wesley Ivan Hurt, Jul 14 2019

Keywords

Links

Crossrefs

Cf. A008683, A326626, A326627, A326629, A326630, A326631, A326632, A326633, A326634, A326635, A326636, A326637.

Programs

  • Mathematica
    Table[Sum[Sum[Sum[Sum[Sum[Sum[Sum[Sum[Sum[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}]
Table[Total[Select[IntegerPartitions[n,{10}],AllTrue[#,SquareFreeQ]&][[;;,-1]]],{n,0,60}] (* Harvey P. Dale, Sep 16 2024 *)

Formula

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 * r, where mu is the Möbius function (A008683).
a(n) = A326627(n) - A326629(n) - A326630(n) - A326631(n) - A326632(n) - A326633(n) - A326634(n) - A326635(n) - A326636(n) - A326637(n).

A326629 Sum of the ninth largest parts of the partitions of n into 10 squarefree parts.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 2, 4, 5, 8, 9, 13, 17, 25, 28, 38, 46, 61, 71, 90, 105, 137, 156, 194, 223, 278, 314, 381, 435, 526, 594, 706, 798, 951, 1064, 1246, 1398, 1641, 1822, 2117, 2360, 2735, 3030, 3480, 3856, 4427, 4884, 5555, 6140, 6984
Offset: 0

Views

Author

Wesley Ivan Hurt, Jul 14 2019

Keywords

Links

Crossrefs

Cf. A008683, A326626, A326627, A326628, A326630, A326631, A326632, A326633, A326634, A326635, A326636, A326637.

Programs

  • Mathematica
    Table[Sum[Sum[Sum[Sum[Sum[Sum[Sum[Sum[Sum[q * 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}]

Formula

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 * q, where mu is the Möbius function (A008683).
a(n) = A326627(n) - A326628(n) - A326630(n) - A326631(n) - A326632(n) - A326633(n) - A326634(n) - A326635(n) - A326636(n) - A326637(n).

A326630 Sum of the eighth largest parts in the partitions of n into 10 squarefree parts.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 2, 4, 5, 8, 9, 14, 18, 26, 30, 41, 50, 66, 77, 100, 117, 152, 174, 219, 252, 314, 357, 436, 499, 605, 685, 820, 929, 1109, 1243, 1469, 1650, 1947, 2169, 2536, 2833, 3297, 3663, 4235, 4707, 5424, 6000, 6867, 7604, 8684
Offset: 0

Views

Author

Wesley Ivan Hurt, Jul 14 2019

Keywords

Links

Crossrefs

Cf. A008683, A005117, A326626, A326627, A326628, A326629, A326631, A326632, A326633, A326634, A326635, A326636, A326637.

Programs

  • Mathematica
    Table[Select[IntegerPartitions[n,{10}],AllTrue[#,SquareFreeQ]&][[All,8]]//Total,{n,0,60}] (* Harvey P. Dale, Apr 19 2020 *)

Formula

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 * p, where mu is the Möbius function (A008683).
a(n) = A326627(n) - A326628(n) - A326629(n) - A326631(n) - A326632(n) - A326633(n) - A326634(n) - A326635(n) - A326636(n) - A326637(n).

A326631 Sum of the seventh largest parts of the partitions of n into 10 squarefree parts.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 2, 4, 5, 8, 10, 15, 19, 28, 33, 45, 55, 73, 86, 113, 132, 171, 198, 250, 288, 359, 411, 504, 578, 700, 798, 961, 1090, 1306, 1477, 1752, 1976, 2338, 2625, 3080, 3457, 4029, 4508, 5228, 5831, 6733, 7490, 8594, 9547
Offset: 0

Views

Author

Wesley Ivan Hurt, Jul 14 2019

Keywords

Links

Crossrefs

Cf. A008683, A326626, A326627, A326628, A326629, A326630, A326632, A326633, A326634, A326635, A326636, A326637.

Programs

  • Mathematica
    Table[Sum[Sum[Sum[Sum[Sum[Sum[Sum[Sum[Sum[o * 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}]

Formula

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 * o, where mu is the Möbius function (A008683).
a(n) = A326627(n) - A326628(n) - A326629(n) - A326630(n) - A326632(n) - A326633(n) - A326634(n) - A326635(n) - A326636(n) - A326637(n).

A326632 Sum of the sixth largest parts of the partitions of n into 10 squarefree parts.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 2, 4, 5, 9, 11, 16, 21, 31, 37, 51, 61, 83, 99, 129, 150, 197, 229, 288, 332, 417, 479, 589, 676, 830, 950, 1150, 1310, 1588, 1802, 2148, 2431, 2897, 3264, 3843, 4322, 5067, 5684, 6604, 7380, 8557, 9538, 10961, 12198
Offset: 0

Views

Author

Wesley Ivan Hurt, Jul 14 2019

Keywords

Links

Crossrefs

Cf. A008683, A326626, A326627, A326628, A326629, A326630, A326631, A326633, A326634, A326635, A326636, A326637.

Programs

  • Mathematica
    Table[Sum[Sum[Sum[Sum[Sum[Sum[Sum[Sum[Sum[m * 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}]

Formula

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 * m, where mu is the Möbius function (A008683).
a(n) = A326627(n) - A326628(n) - A326629(n) - A326630(n) - A326631(n) - A326633(n) - A326634(n) - A326635(n) - A326636(n) - A326637(n).

A326633 Sum of the fifth largest parts of the partitions of n into 10 squarefree parts.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 2, 4, 6, 10, 12, 18, 24, 36, 42, 58, 71, 97, 114, 149, 176, 230, 266, 338, 394, 498, 575, 714, 832, 1028, 1183, 1439, 1656, 2011, 2290, 2735, 3115, 3711, 4195, 4936, 5574, 6533, 7335, 8523, 9549, 11060, 12334, 14162
Offset: 0

Views

Author

Wesley Ivan Hurt, Jul 14 2019

Keywords

Links

Crossrefs

Cf. A008683, A326626, A326627, A326628, A326629, A326630, A326631, A326632, A326634, A326635, A326636, A326637.

Programs

  • Mathematica
    Table[Total[Select[IntegerPartitions[n,{10}],AllTrue[#,SquareFreeQ]&][[All,5]]],{n,0,60}] (* Harvey P. Dale, Sep 29 2021 *)

Formula

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 * l, where mu is the Möbius function (A008683).
a(n) = A326627(n) - A326628(n) - A326629(n) - A326630(n) - A326631(n) - A326632(n) - A326634(n) - A326635(n) - A326636(n) - A326637(n).

A326635 Sum of the third largest parts of the partitions of n into 10 squarefree parts.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 3, 6, 8, 14, 17, 27, 34, 50, 59, 86, 105, 145, 176, 238, 286, 378, 451, 584, 690, 876, 1022, 1280, 1487, 1824, 2104, 2557, 2932, 3536, 4030, 4803, 5463, 6478, 7327, 8633, 9751, 11420, 12854, 14985, 16822, 19536, 21874
Offset: 0

Views

Author

Wesley Ivan Hurt, Jul 14 2019

Keywords

Links

Crossrefs

Cf. A008683, A326626, A326627, A326628, A326629, A326630, A326631, A326632, A326633, A326634, A326636, A326637.

Programs

  • Mathematica
    Table[Sum[Sum[Sum[Sum[Sum[Sum[Sum[Sum[Sum[j * 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}]

Formula

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 * j, 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) - A326636(n) - A326637(n).

A326636 Sum of the second largest parts of the partitions of n into 10 squarefree parts.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 3, 4, 8, 10, 18, 22, 36, 45, 72, 88, 127, 153, 215, 263, 351, 418, 555, 658, 843, 984, 1252, 1460, 1825, 2118, 2623, 3029, 3697, 4248, 5168, 5914, 7101, 8088, 9676, 10960, 12974, 14647, 17246, 19396, 22653, 25384, 29527
Offset: 0

Views

Author

Wesley Ivan Hurt, Jul 14 2019

Keywords

Links

Crossrefs

Cf. A008683, A326626, A326627, A326628, A326629, A326630, A326631, A326632, A326633, A326634, A326635, A326637.

Programs

  • Mathematica
    Table[Total[Select[IntegerPartitions[n,{10}],AllTrue[#,SquareFreeQ]&][[All,2]]],{n,0,55}] (* Harvey P. Dale, Jan 03 2023 *)

Formula

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 * i, 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) - A326637(n).
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