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 1-4 of 4 results.

A308783 Sum of all the parts in the partitions of n into 4 squarefree parts.

Original entry on oeis.org

0, 0, 0, 0, 4, 5, 12, 14, 32, 36, 60, 66, 96, 104, 154, 165, 240, 255, 342, 380, 500, 504, 660, 690, 888, 900, 1144, 1161, 1484, 1508, 1800, 1860, 2272, 2277, 2720, 2800, 3348, 3404, 4028, 4056, 4880, 4879, 5670, 5762, 6820, 6840, 7912, 8084, 9312, 9408
Offset: 0

Views

Author

Wesley Ivan Hurt, Jun 24 2019

Keywords

Links

Crossrefs

Cf. A008683, A308767, A308762, A308768, A308769, A308770.

Programs

  • Mathematica
    Table[n*Sum[Sum[Sum[MoebiusMu[k]^2*MoebiusMu[j]^2*MoebiusMu[i]^2* MoebiusMu[n - i - j - k]^2, {i, j, Floor[(n - j - k)/2]}], {j, k, Floor[(n - k)/3]}], {k, Floor[n/4]}], {n, 0, 50}]
Table[Total[Flatten[Select[IntegerPartitions[n,{4}],AllTrue[#,SquareFreeQ]&]]],{n,0,50}] (* Harvey P. Dale, Aug 14 2022 *)

Formula

a(n) = n * Sum_{k=1..floor(n/4)} Sum_{j=k..floor((n-k)/3)} Sum_{i=j..floor((n-j-k)/2)} mu(k)^2 * mu(j)^2 * mu(i)^2 * mu(n-i-j-k)^2, where mu is the Möbius function (A008683).
a(n) = n * A308767(n).
a(n) = A308768(n) + A308762(n) + A308769(n) + A308770(n).

A308762 Sum of the third largest parts of the partitions of n into 4 squarefree parts.

Original entry on oeis.org

0, 0, 0, 0, 1, 1, 2, 3, 6, 6, 10, 11, 16, 16, 22, 23, 35, 38, 51, 57, 75, 76, 94, 99, 125, 128, 158, 162, 208, 209, 242, 251, 311, 317, 376, 390, 467, 478, 548, 553, 672, 682, 784, 801, 957, 957, 1096, 1101, 1284, 1294, 1471, 1469, 1725, 1717, 1917, 1918
Offset: 0

Views

Author

Wesley Ivan Hurt, Jun 23 2019

Keywords

Links

Crossrefs

Cf. A008683, A308767, A308768, A308769, A308770, A308783.

Programs

  • Mathematica
    Table[Sum[Sum[Sum[j * MoebiusMu[k]^2*MoebiusMu[j]^2*MoebiusMu[i]^2* MoebiusMu[n - i - j - k]^2, {i, j, Floor[(n - j - k)/2]}], {j, k, Floor[(n - k)/3]}], {k, Floor[n/4]}], {n, 0, 100}]

Formula

a(n) = Sum_{k=1..floor(n/4)} Sum_{j=k..floor((n-k)/3)} Sum_{i=j..floor((n-j-k)/2)} mu(k)^2 * mu(j)^2 * mu(i)^2 * mu(n-i-j-k)^2 * j, where mu is the Möbius function (A008683).
a(n) = A308783(n) - A308768(n) - A308769(n) - A308770(n).

A308768 Sum of the smallest parts of the partitions of n into 4 squarefree parts.

Original entry on oeis.org

0, 0, 0, 0, 1, 1, 2, 2, 5, 5, 7, 8, 12, 11, 16, 16, 23, 23, 30, 32, 44, 43, 56, 57, 72, 72, 90, 87, 114, 112, 135, 137, 169, 164, 197, 196, 233, 238, 282, 276, 337, 332, 381, 378, 454, 447, 525, 523, 606, 609, 698, 678, 800, 799, 907, 895, 1050, 1022, 1157
Offset: 0

Views

Author

Wesley Ivan Hurt, Jun 23 2019

Keywords

Links

Crossrefs

Cf. A008683, A308762, A308767, A308769, A308770, A308783.

Programs

  • Mathematica
    Table[Sum[Sum[Sum[k * MoebiusMu[k]^2*MoebiusMu[j]^2*MoebiusMu[i]^2* MoebiusMu[n - i - j - k]^2, {i, j, Floor[(n - j - k)/2]}], {j, k, Floor[(n - k)/3]}], {k, Floor[n/4]}], {n, 0, 100}]

Formula

a(n) = Sum_{k=1..floor(n/4)} Sum_{j=k..floor((n-k)/3)} Sum_{i=j..floor((n-j-k)/2)} mu(k)^2 * mu(j)^2 * mu(i)^2 * mu(n-i-j-k)^2 * k, where mu is the Möbius function (A008683).
a(n) = A308783(n) - A308762(n) - A308769(n) - A308770(n).

A308770 Sum of the largest parts of the partitions of n into 4 squarefree parts.

Original entry on oeis.org

0, 0, 0, 0, 1, 2, 5, 5, 13, 17, 29, 32, 44, 52, 75, 81, 118, 130, 176, 198, 261, 262, 351, 362, 470, 478, 617, 621, 787, 801, 951, 978, 1182, 1184, 1413, 1469, 1747, 1789, 2123, 2160, 2574, 2593, 3012, 3093, 3644, 3679, 4245, 4384, 5024, 5097, 5738, 5891
Offset: 0

Views

Author

Wesley Ivan Hurt, Jun 23 2019

Keywords

Links

Crossrefs

Cf. A008683, A308762, A308767, A308768, A308769, A308783.

Programs

  • Mathematica
    Table[Sum[Sum[Sum[(n - i - j - k) * MoebiusMu[k]^2*MoebiusMu[j]^2* MoebiusMu[i]^2*MoebiusMu[n - i - j - k]^2, {i, j, Floor[(n - j - k)/2]}], {j, k, Floor[(n - k)/3]}], {k, Floor[n/4]}], {n, 0, 100}]
Table[Total[Select[IntegerPartitions[n,{4}],AllTrue[#,SquareFreeQ]&][[;;,1]]],{n,0,60}] (* Harvey P. Dale, Oct 04 2023 *)

Formula

a(n) = Sum_{k=1..floor(n/4)} Sum_{j=k..floor((n-k)/3)} Sum_{i=j..floor((n-j-k)/2)} mu(k)^2 * mu(j)^2 * mu(i)^2 * mu(n-i-j-k)^2 * (n-i-j-k) , where mu is the Möbius function (A008683).
a(n) = A308783(n) - A308768(n) - A308762(n) - A308769(n).
Showing 1-4 of 4 results.

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