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 1-8 of 8 results.

A341064 Number of ways to write n as an ordered sum of 4 squarefree numbers.

Original entry on oeis.org

1, 4, 10, 16, 23, 32, 50, 68, 83, 92, 116, 148, 178, 192, 224, 276, 335, 360, 400, 460, 547, 580, 634, 704, 821, 868, 938, 1024, 1162, 1212, 1288, 1392, 1572, 1628, 1742, 1876, 2123, 2172, 2308, 2460, 2761, 2820, 2964, 3176, 3550, 3628, 3778, 4028, 4481, 4528, 4686, 4932, 5513, 5564
Offset: 4

Views

Author

Ilya Gutkovskiy, Feb 04 2021

Keywords

Crossrefs

Cf. A005117, A008683, A008966, A098235, A280210, A308767, A341065, A341066, A341067, A341068, A341069, A341070.

Programs

  • Maple
    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, 4):
seq(a(n), n=4..57);  # Alois P. Heinz, Feb 04 2021
  • Mathematica
    nmax = 57; CoefficientList[Series[Sum[MoebiusMu[k]^2 x^k, {k, 1, nmax}]^4, {x, 0, nmax}], x] // Drop[#, 4] &

Formula

G.f.: (Sum_{k>=1} mu(k)^2 * x^k)^4.

A341073 Number of partitions of n into 4 distinct squarefree parts.

Original entry on oeis.org

1, 1, 1, 1, 2, 4, 3, 2, 5, 7, 8, 7, 11, 13, 15, 13, 17, 20, 23, 21, 28, 33, 34, 32, 40, 44, 47, 44, 55, 63, 66, 62, 75, 84, 87, 81, 98, 110, 115, 109, 127, 144, 148, 140, 159, 180, 186, 177, 199, 220, 231, 217, 241, 264, 275, 262, 290, 317, 325, 314, 343, 376, 382, 368, 403
Offset: 11

Views

Author

Ilya Gutkovskiy, Feb 04 2021

Keywords

Links

Crossrefs

Cf. A005117, A008966, A098236, A307835, A308767, A341064, A341074, A341075, A341095, A341096, A341097, A341098.

Programs

  • Maple
    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, 4):
seq(a(n), n=11..75);  # Alois P. Heinz, Feb 04 2021
  • Mathematica
    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, 4];
Table[a[n], {n, 11, 75}] (* Jean-François Alcover, Jul 14 2021, after Alois P. Heinz *)

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).

A308769 Sum of the second largest parts of the partitions of n into 4 squarefree parts.

Original entry on oeis.org

0, 0, 0, 0, 1, 1, 3, 4, 8, 8, 14, 15, 24, 25, 41, 45, 64, 64, 85, 93, 120, 123, 159, 172, 221, 222, 279, 291, 375, 386, 472, 494, 610, 612, 734, 745, 901, 899, 1075, 1067, 1297, 1272, 1493, 1490, 1765, 1757, 2046, 2076, 2398, 2408, 2743, 2774, 3187, 3177
Offset: 0

Views

Author

Wesley Ivan Hurt, Jun 23 2019

Keywords

Links

Crossrefs

Cf. A008683, A308762, A308767, A308768, A308770, A308783.

Programs

  • Mathematica
    Table[Sum[Sum[Sum[i * 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 * i, where mu(n) is the Möbius function (A008683).
a(n) = A308783(n) - A308768(n) - A308762(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).

A347655 Number of partitions of n into at most 4 squarefree parts.

Original entry on oeis.org

1, 1, 2, 3, 4, 5, 7, 8, 10, 11, 13, 14, 17, 19, 22, 24, 28, 31, 34, 37, 42, 44, 50, 53, 59, 61, 69, 71, 80, 82, 90, 93, 103, 106, 117, 121, 134, 137, 150, 154, 169, 173, 188, 194, 212, 216, 235, 240, 259, 264, 284, 288, 310, 314, 337, 342, 368, 370, 398, 403, 432
Offset: 0

Views

Author

Ilya Gutkovskiy, Sep 09 2021

Keywords

Crossrefs

Cf. A005117, A073576, A308767, A347648, A347649, A347656, A347657.

Programs

Showing 1-8 of 8 results.

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