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 1-7 of 7 results.

A308902 Number of partitions of n into 6 squarefree parts.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 1, 2, 2, 4, 5, 8, 8, 11, 13, 18, 19, 25, 27, 36, 39, 48, 52, 66, 70, 85, 91, 111, 117, 139, 148, 176, 185, 214, 227, 266, 278, 318, 336, 387, 405, 459, 482, 550, 574, 644, 676, 764, 796, 885, 929, 1038, 1082, 1194, 1247, 1385, 1440, 1580
Offset: 0

Views

Author

Wesley Ivan Hurt, Jun 29 2019

Keywords

Links

Crossrefs

Cf. A008683, A308903, A308906, A308907, A308908, A308909, A308910, A308911.

Programs

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

Formula

a(n) = Sum_{m=1..floor(n/6)} Sum_{l=m..floor((n-m)/5)} Sum_{k=l..floor((n-l-m)/4)} Sum_{j=k..floor((n-k-l-m)/3)} Sum_{i=j..floor((n-j-k-l-m)/2)} mu(m)^2 * mu(l)^2 * mu(k)^2 * mu(j)^2 * mu(i)^2 * mu(n-i-k-j-l-m)^2, where mu is the Möbius function (A008683).
a(n) = A308903(n)/n.

A308903 Sum of all the parts in the partitions of n into 6 squarefree parts.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 6, 7, 16, 18, 40, 55, 96, 104, 154, 195, 288, 323, 450, 513, 720, 819, 1056, 1196, 1584, 1750, 2210, 2457, 3108, 3393, 4170, 4588, 5632, 6105, 7276, 7945, 9576, 10286, 12084, 13104, 15480, 16605, 19278, 20726, 24200, 25830, 29624, 31772
Offset: 0

Views

Author

Wesley Ivan Hurt, Jun 29 2019

Keywords

Links

Crossrefs

Cf. A008683, A308902, A308906, A308907, A308908, A308909, A308910, A308911.

Programs

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

Formula

a(n) = n * Sum_{m=1..floor(n/6)} Sum_{l=m..floor((n-m)/5)} Sum_{k=l..floor((n-l-m)/4)} Sum_{j=k..floor((n-k-l-m)/3)} Sum_{i=j..floor((n-j-k-l-m)/2)} mu(m)^2 * mu(l)^2 * mu(k)^2 * mu(j)^2 * mu(i)^2 * mu(n-i-k-j-l-m)^2, where mu is the Möbius function (A008683).
a(n) = n * A308902(n).
a(n) = A308906(n) + A308907(n) + A308908(n) + A308909(n) + A308910(n) + A308911(n).

A308906 Sum of the smallest parts in the partitions of n into 6 squarefree parts.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 1, 2, 2, 4, 5, 9, 9, 12, 15, 21, 23, 31, 32, 45, 50, 61, 66, 87, 94, 114, 123, 154, 165, 199, 212, 261, 276, 323, 345, 418, 438, 507, 538, 637, 672, 771, 810, 947, 999, 1130, 1192, 1381, 1445, 1625, 1716, 1955, 2045, 2289, 2399, 2720
Offset: 0

Views

Author

Wesley Ivan Hurt, Jun 29 2019

Keywords

Links

Crossrefs

Cf. A008683, A308902, A308903, A308907, A308908, A308909, A308910, A308911.

Programs

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

Formula

a(n) = Sum_{m=1..floor(n/6)} Sum_{l=m..floor((n-m)/5)} Sum_{k=l..floor((n-l-m)/4)} Sum_{j=k..floor((n-k-l-m)/3)} Sum_{i=j..floor((n-j-k-l-m)/2)} mu(m)^2 * mu(l)^2 * mu(k)^2 * mu(j)^2 * mu(i)^2 * mu(n-i-k-j-l-m)^2 * m, where mu is the Möbius function (A008683).
a(n) = A308903(n) - A308907(n) - A308908(n) - A308909(n) - A308910(n) - A308911(n).

A308907 Sum of the fifth largest parts in the partitions of n into 6 squarefree parts.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 1, 2, 2, 4, 6, 10, 10, 14, 18, 26, 28, 37, 41, 57, 62, 77, 87, 113, 122, 152, 170, 213, 230, 279, 307, 376, 402, 471, 516, 622, 661, 768, 830, 978, 1041, 1194, 1282, 1492, 1586, 1804, 1932, 2217, 2340, 2632, 2815, 3195, 3380, 3780, 4026
Offset: 0

Views

Author

Wesley Ivan Hurt, Jun 29 2019

Keywords

Links

Crossrefs

Cf. A008683, A308902, A308903, A308906, A308908, A308909, A308910, A308911.

Programs

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

Formula

a(n) = Sum_{m=1..floor(n/6)} Sum_{l=m..floor((n-m)/5)} Sum_{k=l..floor((n-l-m)/4)} Sum_{j=k..floor((n-k-l-m)/3)} Sum_{i=j..floor((n-j-k-l-m)/2)} mu(m)^2 * mu(l)^2 * mu(k)^2 * mu(j)^2 * mu(i)^2 * mu(n-i-k-j-l-m)^2 * l, where mu is the Möbius function (A008683).
a(n) = A308903(n) - A308906(n) - A308908(n) - A308909(n) - A308910(n) - A308911(n).

A308908 Sum of the fourth largest parts in the partitions of n into 6 squarefree parts.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 1, 2, 2, 5, 7, 11, 12, 18, 22, 32, 34, 47, 52, 71, 78, 102, 116, 154, 170, 217, 243, 305, 329, 406, 445, 546, 587, 702, 768, 921, 982, 1147, 1240, 1459, 1562, 1811, 1948, 2260, 2401, 2748, 2943, 3387, 3596, 4087, 4381, 4987, 5288, 5959
Offset: 0

Views

Author

Wesley Ivan Hurt, Jun 29 2019

Keywords

Links

Crossrefs

Cf. A008683, A308902, A308903, A308906, A308907, A308909, A308910, A308911.

Programs

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

Formula

a(n) = Sum_{m=1..floor(n/6)} Sum_{l=m..floor((n-m)/5)} Sum_{k=l..floor((n-l-m)/4)} Sum_{j=k..floor((n-k-l-m)/3)} Sum_{i=j..floor((n-j-k-l-m)/2)} mu(m)^2 * mu(l)^2 * mu(k)^2 * mu(j)^2 * mu(i)^2 * mu(n-i-k-j-l-m)^2 * k, where mu is the Möbius function (A008683).
a(n) = A308903(n) - A308906(n) - A308907(n) - A308909(n) - A308910(n) - A308911(n).

A308910 Sum of the second largest parts in the partitions of n into 6 squarefree parts.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 1, 3, 4, 8, 10, 18, 20, 32, 38, 60, 70, 100, 112, 157, 181, 231, 259, 341, 382, 479, 531, 672, 743, 917, 1013, 1253, 1378, 1658, 1819, 2205, 2392, 2832, 3065, 3638, 3909, 4572, 4890, 5726, 6104, 7027, 7495, 8656, 9187, 10455, 11130
Offset: 0

Views

Author

Wesley Ivan Hurt, Jun 29 2019

Keywords

Links

Crossrefs

Cf. A008683, A308902, A308903, A308906, A308907, A308908, A308909, A308911.

Programs

  • Mathematica
    Table[Sum[Sum[Sum[Sum[Sum[i*MoebiusMu[i]^2*MoebiusMu[j]^2*MoebiusMu[k]^2* MoebiusMu[l]^2*MoebiusMu[m]^2*MoebiusMu[n - i - j - k - l - m]^2, {i, j, Floor[(n - j - k - l - m)/2]}], {j, k, Floor[(n - k - l - m)/3]}], {k, l, Floor[(n - l - m)/4]}], {l, m, Floor[(n - m)/5]}], {m, Floor[n/6]}], {n, 0, 50}]
Table[Total[Select[IntegerPartitions[n,{6}],AllTrue[#,SquareFreeQ]&][[;;,2]]],{n,0,60}] (* Harvey P. Dale, Jun 16 2024 *)

Formula

a(n) = Sum_{m=1..floor(n/6)} Sum_{l=m..floor((n-m)/5)} Sum_{k=l..floor((n-l-m)/4)} Sum_{j=k..floor((n-k-l-m)/3)} Sum_{i=j..floor((n-j-k-l-m)/2)} mu(m)^2 * mu(l)^2 * mu(k)^2 * mu(j)^2 * mu(i)^2 * mu(n-i-k-j-l-m)^2 * i, where mu is the Möbius function (A008683).
a(n) = A308903(n) - A308906(n) - A308907(n) - A308908(n) - A308909(n) - A308911(n).

A308911 Sum of the largest parts in the partitions of n into 6 squarefree parts.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 2, 5, 5, 13, 19, 34, 38, 55, 74, 110, 125, 173, 206, 292, 333, 433, 493, 662, 729, 929, 1034, 1323, 1441, 1770, 1955, 2403, 2598, 3096, 3376, 4066, 4360, 5121, 5566, 6584, 7064, 8183, 8832, 10326, 11021, 12626, 13592, 15701, 16743, 18957
Offset: 0

Views

Author

Wesley Ivan Hurt, Jun 29 2019

Keywords

Links

Crossrefs

Cf. A008683, A308902, A308903, A308906, A308907, A308908, A308909, A308910.

Programs

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

Formula

a(n) = Sum_{m=1..floor(n/6)} Sum_{l=m..floor((n-m)/5)} Sum_{k=l..floor((n-l-m)/4)} Sum_{j=k..floor((n-k-l-m)/3)} Sum_{i=j..floor((n-j-k-l-m)/2)} mu(m)^2 * mu(l)^2 * mu(k)^2 * mu(j)^2 * mu(i)^2 * mu(n-i-k-j-l-m)^2 * (n-i-j-k-l-m), where mu is the Möbius function (A008683).
a(n) = A308903(n) - A308906(n) - A308907(n) - A308908(n) - A308909(n) - A308910(n).
Showing 1-7 of 7 results.

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