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.

A308843 Sum of the third largest parts in the partitions of n into 5 squarefree parts.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 1, 2, 3, 6, 8, 12, 13, 21, 23, 32, 33, 49, 56, 77, 86, 117, 130, 162, 174, 223, 239, 295, 312, 391, 418, 497, 520, 631, 675, 801, 844, 1009, 1072, 1247, 1306, 1537, 1628, 1890, 1972, 2312, 2425, 2786, 2889, 3325, 3472, 3955, 4089, 4671, 4851, 5474
Offset: 0

Views

Author

Wesley Ivan Hurt, Jun 28 2019

Keywords

Comments

Conjecture: a(4*k + 3) < a(4*k + 4) for 4*k + 3 >= 195. This conjecture holds for all terms in the b-file. - David A. Corneth, Sep 16 2019

Links

Crossrefs

Cf. A008683, A308839, A308840, A308841, A308842, A308844, A308845.

Programs

  • Mathematica
    Table[Sum[Sum[Sum[Sum[j * MoebiusMu[l]^2*MoebiusMu[k]^2*MoebiusMu[j]^2 *MoebiusMu[i]^2*MoebiusMu[n - i - j - k - l]^2, {i, j, Floor[(n - j - k - l)/2]}], {j, k, Floor[(n - k - l)/3]}], {k, l, Floor[(n - l)/4]}], {l, Floor[n/5]}], {n, 0, 50}]
Table[Total[Select[IntegerPartitions[n,{5}],AllTrue[#,SquareFreeQ]&][[All,3]]],{n,0,60}] (* Harvey P. Dale, Dec 26 2022 *)

Formula

a(n) = Sum_{l=1..floor(n/5)} Sum_{k=l..floor((n-l)/4)} Sum_{j=k..floor((n-k-l)/3)} Sum_{i=j..floor((n-j-k-l)/2)} mu(l)^2 * mu(k)^2 * mu(j)^2 * mu(i)^2 * mu(n-i-j-k-l)^2 * j, where mu is the Möbius function (A008683).
a(n) = A308839(n) - A308841(n) - A308842(n) - A308844(n) - A308845(n).

Extensions

a(54)..a(55) from David A. Corneth, Sep 16 2019

A308841 Sum of the smallest parts in the partitions of n into 5 squarefree parts.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 1, 2, 2, 4, 6, 8, 8, 12, 14, 20, 19, 26, 30, 39, 39, 52, 57, 71, 72, 95, 100, 123, 125, 155, 166, 198, 200, 242, 256, 304, 306, 366, 383, 445, 453, 533, 556, 642, 652, 762, 786, 898, 914, 1048, 1091, 1236, 1261, 1434, 1487, 1671, 1695, 1919
Offset: 0

Views

Author

Wesley Ivan Hurt, Jun 28 2019

Keywords

Examples

			The partitions of n into 5 parts for n = 10, 11, ..
                                                       1+1+1+1+10
                                                        1+1+1+2+9
                                                        1+1+1+3+8
                                                        1+1+1+4+7
                                                        1+1+1+5+6
                                            1+1+1+1+9   1+1+2+2+8
                                            1+1+1+2+8   1+1+2+3+7
                                            1+1+1+3+7   1+1+2+4+6
                                            1+1+1+4+6   1+1+2+5+5
                                            1+1+1+5+5   1+1+3+3+6
                                1+1+1+1+8   1+1+2+2+7   1+1+3+4+5
                                1+1+1+2+7   1+1+2+3+6   1+1+4+4+4
                                1+1+1+3+6   1+1+2+4+5   1+2+2+2+7
                    1+1+1+1+7   1+1+1+4+5   1+1+3+3+5   1+2+2+3+6
                    1+1+1+2+6   1+1+2+2+6   1+1+3+4+4   1+2+2+4+5
                    1+1+1+3+5   1+1+2+3+5   1+2+2+2+6   1+2+3+3+5
        1+1+1+1+6   1+1+1+4+4   1+1+2+4+4   1+2+2+3+5   1+2+3+4+4
        1+1+1+2+5   1+1+2+2+5   1+1+3+3+4   1+2+2+4+4   1+3+3+3+4
        1+1+1+3+4   1+1+2+3+4   1+2+2+2+5   1+2+3+3+4   2+2+2+2+6
        1+1+2+2+4   1+1+3+3+3   1+2+2+3+4   1+3+3+3+3   2+2+2+3+5
        1+1+2+3+3   1+2+2+2+4   1+2+3+3+3   2+2+2+2+5   2+2+2+4+4
        1+2+2+2+3   1+2+2+3+3   2+2+2+2+4   2+2+2+3+4   2+2+3+3+4
        2+2+2+2+2   2+2+2+2+3   2+2+2+3+3   2+2+3+3+3   2+3+3+3+3
--------------------------------------------------------------------------
  n  |     10          11          12          13          14        ...
--------------------------------------------------------------------------
a(n) |      6           8           8          12          14        ...
--------------------------------------------------------------------------
- _Wesley Ivan Hurt_, Sep 16 2019

Links

Crossrefs

Cf. A008683, A308839, A308840, A308842, A308843, A308844, A308845.

Programs

  • Mathematica
    Table[Sum[Sum[Sum[Sum[l * MoebiusMu[l]^2*MoebiusMu[k]^2*MoebiusMu[j]^2* MoebiusMu[i]^2*MoebiusMu[n - i - j - k - l]^2, {i, j, Floor[(n - j - k - l)/2]}], {j, k, Floor[(n - k - l)/3]}], {k, l, Floor[(n - l)/4]}], {l, Floor[n/5]}], {n, 0, 50}]

Formula

a(n) = Sum_{l=1..floor(n/5)} Sum_{k=l..floor((n-l)/4)} Sum_{j=k..floor((n-k-l)/3)} Sum_{i=j..floor((n-j-k-l)/2)} mu(l)^2 * mu(k)^2 * mu(j)^2 * mu(i)^2 * mu(n-i-j-k-l)^2 * l, where mu is the Möbius function (A008683).
a(n) = A308839(n) - A308842(n) - A308843(n) - A308844(n) - A308845(n).

A308844 Sum of the second largest parts in the partitions of n into 5 squarefree parts.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 1, 3, 4, 8, 10, 16, 18, 29, 33, 52, 59, 83, 93, 125, 138, 178, 196, 252, 275, 350, 380, 471, 506, 634, 689, 839, 901, 1096, 1176, 1405, 1484, 1767, 1861, 2199, 2294, 2695, 2823, 3281, 3388, 3941, 4101, 4714, 4901, 5607, 5843, 6643, 6893
Offset: 0

Views

Author

Wesley Ivan Hurt, Jun 28 2019

Keywords

Examples

			The partitions of n into 5 parts for n = 10, 11, ..
                                                       1+1+1+1+10
                                                        1+1+1+2+9
                                                        1+1+1+3+8
                                                        1+1+1+4+7
                                                        1+1+1+5+6
                                            1+1+1+1+9   1+1+2+2+8
                                            1+1+1+2+8   1+1+2+3+7
                                            1+1+1+3+7   1+1+2+4+6
                                            1+1+1+4+6   1+1+2+5+5
                                            1+1+1+5+5   1+1+3+3+6
                                1+1+1+1+8   1+1+2+2+7   1+1+3+4+5
                                1+1+1+2+7   1+1+2+3+6   1+1+4+4+4
                                1+1+1+3+6   1+1+2+4+5   1+2+2+2+7
                    1+1+1+1+7   1+1+1+4+5   1+1+3+3+5   1+2+2+3+6
                    1+1+1+2+6   1+1+2+2+6   1+1+3+4+4   1+2+2+4+5
                    1+1+1+3+5   1+1+2+3+5   1+2+2+2+6   1+2+3+3+5
        1+1+1+1+6   1+1+1+4+4   1+1+2+4+4   1+2+2+3+5   1+2+3+4+4
        1+1+1+2+5   1+1+2+2+5   1+1+3+3+4   1+2+2+4+4   1+3+3+3+4
        1+1+1+3+4   1+1+2+3+4   1+2+2+2+5   1+2+3+3+4   2+2+2+2+6
        1+1+2+2+4   1+1+3+3+3   1+2+2+3+4   1+3+3+3+3   2+2+2+3+5
        1+1+2+3+3   1+2+2+2+4   1+2+3+3+3   2+2+2+2+5   2+2+2+4+4
        1+2+2+2+3   1+2+2+3+3   2+2+2+2+4   2+2+2+3+4   2+2+3+3+4
        2+2+2+2+2   2+2+2+2+3   2+2+2+3+3   2+2+3+3+3   2+3+3+3+3
--------------------------------------------------------------------------
  n  |     10          11          12          13          14        ...
--------------------------------------------------------------------------
a(n) |     10          16          18          29          33        ...
--------------------------------------------------------------------------
- _Wesley Ivan Hurt_, Sep 16 2019

Links

Crossrefs

Cf. A008683, A308839, A308840, A308841, A308842, A308843, A308845.

Programs

  • Mathematica
    Table[Total[Select[IntegerPartitions[n,{5}],AllTrue[#,SquareFreeQ]&][[All,2]]],{n,0,60}] (* Harvey P. Dale, Nov 19 2022 *)

Formula

a(n) = Sum_{l=1..floor(n/5)} Sum_{k=l..floor((n-l)/4)} Sum_{j=k..floor((n-k-l)/3)} Sum_{i=j..floor((n-j-k-l)/2)} mu(l)^2 * mu(k)^2 * mu(j)^2 * mu(i)^2 * mu(n-i-j-k-l)^2 * i, where mu is the Möbius function (A008683).
a(n) = A308839(n) - A308841(n) - A308842(n) - A308843(n) - A308845(n).

A308845 Sum of the largest parts in the partitions of n into 5 squarefree parts.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 2, 5, 5, 13, 19, 32, 35, 52, 66, 96, 104, 147, 177, 240, 263, 356, 392, 514, 543, 712, 763, 972, 1015, 1271, 1370, 1652, 1728, 2084, 2223, 2623, 2750, 3275, 3480, 4087, 4276, 5011, 5312, 6141, 6388, 7443, 7842, 8987, 9405, 10753, 11335
Offset: 0

Views

Author

Wesley Ivan Hurt, Jun 28 2019

Keywords

Examples

			The partitions of n into 5 parts for n = 10, 11, ..
                                                       1+1+1+1+10
                                                        1+1+1+2+9
                                                        1+1+1+3+8
                                                        1+1+1+4+7
                                                        1+1+1+5+6
                                            1+1+1+1+9   1+1+2+2+8
                                            1+1+1+2+8   1+1+2+3+7
                                            1+1+1+3+7   1+1+2+4+6
                                            1+1+1+4+6   1+1+2+5+5
                                            1+1+1+5+5   1+1+3+3+6
                                1+1+1+1+8   1+1+2+2+7   1+1+3+4+5
                                1+1+1+2+7   1+1+2+3+6   1+1+4+4+4
                                1+1+1+3+6   1+1+2+4+5   1+2+2+2+7
                    1+1+1+1+7   1+1+1+4+5   1+1+3+3+5   1+2+2+3+6
                    1+1+1+2+6   1+1+2+2+6   1+1+3+4+4   1+2+2+4+5
                    1+1+1+3+5   1+1+2+3+5   1+2+2+2+6   1+2+3+3+5
        1+1+1+1+6   1+1+1+4+4   1+1+2+4+4   1+2+2+3+5   1+2+3+4+4
        1+1+1+2+5   1+1+2+2+5   1+1+3+3+4   1+2+2+4+4   1+3+3+3+4
        1+1+1+3+4   1+1+2+3+4   1+2+2+2+5   1+2+3+3+4   2+2+2+2+6
        1+1+2+2+4   1+1+3+3+3   1+2+2+3+4   1+3+3+3+3   2+2+2+3+5
        1+1+2+3+3   1+2+2+2+4   1+2+3+3+3   2+2+2+2+5   2+2+2+4+4
        1+2+2+2+3   1+2+2+3+3   2+2+2+2+4   2+2+2+3+4   2+2+3+3+4
        2+2+2+2+2   2+2+2+2+3   2+2+2+3+3   2+2+3+3+3   2+3+3+3+3
--------------------------------------------------------------------------
  n  |     10          11          12          13          14        ...
--------------------------------------------------------------------------
a(n) |     19          32          35          52          66        ...
--------------------------------------------------------------------------
- _Wesley Ivan Hurt_, Sep 16 2019

Links

Crossrefs

Cf. A008683, A308839, A308840, A308841, A308842, A308843, A308844.

Programs

  • Mathematica
    Table[Sum[Sum[Sum[Sum[(n - i - j - k - l) * MoebiusMu[l]^2*MoebiusMu[k]^2*MoebiusMu[j]^2*MoebiusMu[i]^2*MoebiusMu[n - i - j - k - l]^2, {i, j, Floor[(n - j - k - l)/2]}], {j, k, Floor[(n - k - l)/3]}], {k, l, Floor[(n - l)/4]}], {l, Floor[n/5]}], {n, 0, 80}]

Formula

a(n) = Sum_{l=1..floor(n/5)} Sum_{k=l..floor((n-l)/4)} Sum_{j=k..floor((n-k-l)/3)} Sum_{i=j..floor((n-j-k-l)/2)} mu(l)^2 * mu(k)^2 * mu(j)^2 * mu(i)^2 * mu(n-i-j-k-l)^2 * (n-i-j-k-l), where mu is the Möbius function (A008683).
a(n) = A308839(n) - A308841(n) - A308842(n) - A308843(n) - A308844(n).
Showing 1-4 of 4 results.

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