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 1-9 of 9 results.

A326443 Number of partitions of n into 8 squarefree parts.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 2, 4, 5, 8, 9, 13, 15, 21, 24, 32, 36, 47, 53, 68, 75, 94, 105, 130, 143, 174, 192, 231, 254, 301, 330, 389, 424, 495, 539, 626, 678, 781, 847, 970, 1048, 1192, 1287, 1461, 1572, 1772, 1908, 2144, 2301, 2573, 2762, 3079, 3295
Offset: 0

Views

Author

Wesley Ivan Hurt, Jul 06 2019

Keywords

Links

Crossrefs

Cf. A008683, A326444, A326445, A326446, A326447, A326448, A326449, A326450, A326451, A326452.

Programs

Formula

a(n) = Sum_{p=1..floor(n/8)} Sum_{o=p..floor((n-p)/7)} Sum_{m=o..floor((n-o-p)/6)} Sum_{l=m..floor((n-m-o-p)/5)} Sum_{k=l..floor((n-l-m-o-p)/4)} Sum_{j=k..floor((n-k-l-m-o-p)/3)} Sum_{i=j..floor((n-j-k-l-m-o-p)/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)^2, where mu is the Möbius function (A008683).
a(n) = A326444(n)/n for n > 0.

A326444 Sum of all the parts in the partitions of n into 8 squarefree parts.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 8, 9, 20, 22, 48, 65, 112, 135, 208, 255, 378, 456, 640, 756, 1034, 1219, 1632, 1875, 2444, 2835, 3640, 4147, 5220, 5952, 7392, 8382, 10234, 11550, 14004, 15688, 18810, 21021, 25040, 27798, 32802, 36421, 42680, 47160, 54832, 60489
Offset: 0

Views

Author

Wesley Ivan Hurt, Jul 06 2019

Keywords

Links

Crossrefs

Cf. A008683, A326443, A326445, A326446, A326447, A326448, A326449, A326450, A326451, A326452.

Programs

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

Formula

a(n) = n * Sum_{p=1..floor(n/8)} Sum_{o=p..floor((n-p)/7)} Sum_{m=o..floor((n-o-p)/6)} Sum_{l=m..floor((n-m-o-p)/5)} Sum_{k=l..floor((n-l-m-o-p)/4)} Sum_{j=k..floor((n-k-l-m-o-p)/3)} Sum_{i=j..floor((n-j-k-l-m-o-p)/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)^2, where mu is the Möbius function (A008683).
a(n) = n * A326443(n).
a(n) = A326445(n) + A326446(n) + A326447(n) + A326448(n) + A326449(n) + A326450(n) + A326451(n) + A326452(n).

A326445 Sum of the smallest parts of the partitions of n into 8 squarefree parts.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 2, 4, 5, 8, 9, 14, 16, 22, 26, 35, 40, 52, 59, 78, 86, 108, 122, 153, 169, 207, 231, 280, 310, 371, 409, 487, 535, 630, 688, 812, 883, 1028, 1119, 1295, 1409, 1619, 1754, 2014, 2180, 2479, 2679, 3046, 3284, 3707, 3994, 4502
Offset: 0

Views

Author

Wesley Ivan Hurt, Jul 06 2019

Keywords

Links

Crossrefs

Cf. A008683, A326443, A326444, A326446, A326447, A326448, A326449, A326450, A326451, A326452.

Programs

  • Mathematica
    Table[Sum[Sum[Sum[Sum[Sum[Sum[Sum[p * 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]^2, {i, j, Floor[(n - j - k - l - m - o - p)/2]}], {j, k, Floor[(n - k - l - m - o - p)/3]}], {k, l, Floor[(n - l - m - o - p)/4]}], {l, m, Floor[(n - m - o - p)/5]}], {m, o, Floor[(n - o - p)/6]}], {o, p, Floor[(n - p)/7]}], {p, Floor[n/8]}], {n, 0, 50}]
Table[Total[Select[IntegerPartitions[n,{8}],AllTrue[#,SquareFreeQ]&][[;;,-1]]],{n,0,60}] (* Harvey P. Dale, Jul 21 2024 *)

Formula

a(n) = Sum_{p=1..floor(n/8)} Sum_{o=p..floor((n-p)/7)} Sum_{m=o..floor((n-o-p)/6)} Sum_{l=m..floor((n-m-o-p)/5)} Sum_{k=l..floor((n-l-m-o-p)/4)} Sum_{j=k..floor((n-k-l-m-o-p)/3)} Sum_{i=j..floor((n-j-k-l-m-o-p)/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)^2 * p, where mu is the Möbius function (A008683).
a(n) = A326444(n) - A326446(n) - A326447(n) - A326448(n) - A326449(n) - A326450(n) - A326451(n) - A326452(n).

A326446 Sum of the seventh largest parts of the partitions of n into 8 squarefree parts.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 2, 4, 5, 8, 10, 15, 17, 24, 29, 39, 45, 59, 68, 90, 100, 126, 144, 181, 202, 248, 279, 340, 380, 454, 506, 608, 673, 795, 879, 1039, 1140, 1331, 1467, 1704, 1868, 2146, 2353, 2713, 2955, 3366, 3672, 4188, 4547, 5142, 5588
Offset: 0

Views

Author

Wesley Ivan Hurt, Jul 06 2019

Keywords

Links

Crossrefs

Cf. A008683, A326443, A326444, A326445, A326447, A326448, A326449, A326450, A326451, A326452.

Programs

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

Formula

a(n) = Sum_{p=1..floor(n/8)} Sum_{o=p..floor((n-p)/7)} Sum_{m=o..floor((n-o-p)/6)} Sum_{l=m..floor((n-m-o-p)/5)} Sum_{k=l..floor((n-l-m-o-p)/4)} Sum_{j=k..floor((n-k-l-m-o-p)/3)} Sum_{i=j..floor((n-j-k-l-m-o-p)/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)^2 * o, where mu is the Möbius function (A008683).
a(n) = A326444(n) - A326445(n) - A326447(n) - A326448(n) - A326449(n) - A326450(n) - A326451(n) - A326452(n).

A326447 Sum of the sixth largest parts in the partitions of n into 8 squarefree parts.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 2, 4, 5, 9, 11, 16, 19, 27, 33, 45, 51, 69, 80, 105, 117, 150, 172, 216, 242, 300, 339, 416, 466, 568, 636, 768, 852, 1022, 1135, 1348, 1483, 1748, 1934, 2260, 2481, 2876, 3163, 3655, 3993, 4582, 5014, 5735, 6244, 7098, 7732
Offset: 0

Views

Author

Wesley Ivan Hurt, Jul 06 2019

Keywords

Links

Crossrefs

Cf. A008683, A326443, A326444, A326445, A326446, A326448, A326449, A326450, A326451, A326452.

Programs

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

Formula

a(n) = Sum_{p=1..floor(n/8)} Sum_{o=p..floor((n-p)/7)} Sum_{m=o..floor((n-o-p)/6)} Sum_{l=m..floor((n-m-o-p)/5)} Sum_{k=l..floor((n-l-m-o-p)/4)} Sum_{j=k..floor((n-k-l-m-o-p)/3)} Sum_{i=j..floor((n-j-k-l-m-o-p)/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)^2 * m, where mu is the Möbius function (A008683).
a(n) = A326444(n) - A326445(n) - A326446(n) - A326448(n) - A326449(n) - A326450(n) - A326451(n) - A326452(n).

A326448 Sum of the fifth largest parts of the partitions of n into 8 squarefree parts.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 2, 4, 6, 10, 12, 18, 22, 32, 38, 52, 61, 82, 94, 124, 141, 180, 206, 262, 298, 373, 426, 527, 600, 735, 828, 1001, 1124, 1348, 1506, 1790, 1989, 2343, 2603, 3040, 3357, 3893, 4295, 4963, 5450, 6254, 6856, 7824, 8550, 9705
Offset: 0

Views

Author

Wesley Ivan Hurt, Jul 06 2019

Keywords

Links

Crossrefs

Cf. A008683, A326443, A326444, A326445, A326446, A326447, A326449, A326450, A326451, A326452.

Programs

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

Formula

a(n) = Sum_{p=1..floor(n/8)} Sum_{o=p..floor((n-p)/7)} Sum_{m=o..floor((n-o-p)/6)} Sum_{l=m..floor((n-m-o-p)/5)} Sum_{k=l..floor((n-l-m-o-p)/4)} Sum_{j=k..floor((n-k-l-m-o-p)/3)} Sum_{i=j..floor((n-j-k-l-m-o-p)/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)^2 * l, where mu is the Möbius function (A008683).
a(n) = A326444(n) - A326445(n) - A326446(n) - A326447(n) - A326449(n) - A326450(n) - A326451(n) - A326452(n).

A326449 Sum of the fourth largest parts of the partitions of n into 8 squarefree parts.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 2, 5, 7, 11, 14, 22, 26, 38, 45, 63, 73, 98, 113, 152, 174, 227, 264, 342, 394, 499, 570, 712, 810, 993, 1119, 1361, 1528, 1833, 2049, 2433, 2704, 3182, 3530, 4127, 4564, 5289, 5828, 6738, 7399, 8490, 9314, 10656, 11671
Offset: 0

Views

Author

Wesley Ivan Hurt, Jul 06 2019

Keywords

Links

Crossrefs

Cf. A008683, A326443, A326444, A326445, A326446, A326447, A326448, A326450, A326451, A326452.

Programs

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

Formula

a(n) = Sum_{p=1..floor(n/8)} Sum_{o=p..floor((n-p)/7)} Sum_{m=o..floor((n-o-p)/6)} Sum_{l=m..floor((n-m-o-p)/5)} Sum_{k=l..floor((n-l-m-o-p)/4)} Sum_{j=k..floor((n-k-l-m-o-p)/3)} Sum_{i=j..floor((n-j-k-l-m-o-p)/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)^2 * k, where mu is the Möbius function (A008683).
a(n) = A326444(n) - A326445(n) - A326446(n) - A326447(n) - A326448(n) - A326450(n) - A326451(n) - A326452(n).

A326450 Sum of the third largest parts of the partitions of n into 8 squarefree parts.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 3, 6, 8, 14, 17, 27, 32, 46, 55, 79, 93, 128, 153, 208, 245, 319, 375, 483, 556, 697, 799, 993, 1127, 1368, 1547, 1871, 2101, 2507, 2809, 3341, 3725, 4377, 4878, 5722, 6350, 7382, 8179, 9510, 10503, 12106, 13352, 15363, 16888
Offset: 0

Views

Author

Wesley Ivan Hurt, Jul 06 2019

Keywords

Links

Crossrefs

Cf. A008683, A326443, A326444, A326445, A326446, A326447, A326448, A326449, A326451, A326452.

Programs

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

Formula

a(n) = Sum_{p=1..floor(n/8)} Sum_{o=p..floor((n-p)/7)} Sum_{m=o..floor((n-o-p)/6)} Sum_{l=m..floor((n-m-o-p)/5)} Sum_{k=l..floor((n-l-m-o-p)/4)} Sum_{j=k..floor((n-k-l-m-o-p)/3)} Sum_{i=j..floor((n-j-k-l-m-o-p)/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)^2 * j, where mu is the Möbius function (A008683).
a(n) = A326444(n) - A326445(n) - A326446(n) - A326447(n) - A326448(n) - A326449(n) - A326451(n) - A326452(n).

A326451 Sum of the second largest parts of the partitions of n into 8 squarefree parts.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 3, 4, 8, 10, 18, 22, 36, 43, 68, 83, 119, 140, 196, 235, 312, 361, 471, 550, 704, 802, 1008, 1157, 1446, 1643, 2016, 2294, 2798, 3154, 3807, 4285, 5135, 5728, 6797, 7571, 8926, 9880, 11543, 12744, 14827, 16295, 18801, 20645
Offset: 0

Views

Author

Wesley Ivan Hurt, Jul 06 2019

Keywords

Links

Crossrefs

Cf. A008683, A326443, A326444, A326445, A326446, A326447, A326448, A326449, A326450, A326452.

Programs

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

Formula

a(n) = Sum_{p=1..floor(n/8)} Sum_{o=p..floor((n-p)/7)} Sum_{m=o..floor((n-o-p)/6)} Sum_{l=m..floor((n-m-o-p)/5)} Sum_{k=l..floor((n-l-m-o-p)/4)} Sum_{j=k..floor((n-k-l-m-o-p)/3)} Sum_{i=j..floor((n-j-k-l-m-o-p)/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)^2 * i, where mu is the Möbius function (A008683).
a(n) = A326444(n) - A326445(n) - A326446(n) - A326447(n) - A326448(n) - A326449(n) - A326450(n) - A326452(n).
Showing 1-9 of 9 results.

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