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 1-10 of 10 results.

A341066 Number of ways to write n as an ordered sum of 6 squarefree numbers.

Original entry on oeis.org

1, 6, 21, 50, 96, 162, 267, 426, 645, 902, 1218, 1632, 2187, 2826, 3543, 4402, 5547, 6906, 8397, 10032, 12108, 14578, 17298, 20112, 23517, 27534, 32034, 36592, 41892, 48018, 54886, 61758, 69549, 78408, 88365, 98274, 109478, 122058, 136230, 150114, 165759, 183114, 202630, 221484
Offset: 6

Views

Author

Ilya Gutkovskiy, Feb 04 2021

Keywords

Examples

			G.f.: (Sum_{k>=1} mu(k)^2 * x^k)^6.

Crossrefs

Cf. A005117, A008683, A008966, A098235, A280210, A308902, A341064, A341065, 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, 6):
seq(a(n), n=6..49);  # Alois P. Heinz, Feb 04 2021
  • Mathematica
    nmax = 49; CoefficientList[Series[Sum[MoebiusMu[k]^2 x^k, {k, 1, nmax}]^6, {x, 0, nmax}], x] // Drop[#, 6] &

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).

A308909 Sum of the third 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, 3, 6, 8, 14, 15, 23, 28, 39, 43, 62, 70, 98, 115, 152, 175, 227, 253, 319, 356, 441, 485, 599, 656, 793, 864, 1026, 1121, 1344, 1453, 1709, 1865, 2184, 2357, 2747, 2964, 3449, 3719, 4289, 4618, 5330, 5693, 6494, 6956, 7922, 8430
Offset: 0

Views

Author

Wesley Ivan Hurt, Jun 29 2019

Keywords

Links

Crossrefs

Cf. A008683, A308902, A308903, A308906, A308907, A308908, A308910, A308911.

Programs

  • Mathematica
    Table[Total[Select[IntegerPartitions[n,{6}],AllTrue[#,SquareFreeQ]&][[All,3]]],{n,0,60}] (* Harvey P. Dale, Jan 31 2022 *)

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 * j, where mu is the Möbius function (A008683).
a(n) = A308903(n) - A308906(n) - A308907(n) - A308908(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).

A341075 Number of partitions of n into 6 distinct squarefree parts.

Original entry on oeis.org

1, 0, 0, 1, 2, 2, 2, 3, 6, 5, 6, 7, 12, 12, 15, 18, 26, 26, 28, 34, 44, 46, 50, 60, 77, 79, 86, 98, 122, 126, 134, 154, 188, 196, 207, 236, 277, 292, 305, 343, 400, 423, 443, 492, 567, 596, 624, 686, 779, 819, 856, 938, 1052, 1108, 1149, 1255, 1394, 1463, 1515, 1646, 1818
Offset: 24

Views

Author

Ilya Gutkovskiy, Feb 04 2021

Keywords

Crossrefs

Cf. A005117, A008966, A098236, A307835, A308902, A341066, A341073, A341074, 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, 6):
seq(a(n), n=24..84);  # Alois P. Heinz, Feb 04 2021
  • Mathematica
    Table[Length[Select[IntegerPartitions[n,{6}],Length[Union[#]]==6&&AllTrue[ #,SquareFreeQ]&]],{n,24,90}] (* Harvey P. Dale, Jan 16 2022 *)

A347657 Number of partitions of n into at most 6 squarefree parts.

Original entry on oeis.org

1, 1, 2, 3, 4, 6, 9, 11, 14, 17, 22, 26, 32, 37, 44, 52, 61, 70, 81, 92, 107, 120, 137, 153, 174, 192, 217, 238, 268, 292, 326, 354, 394, 426, 471, 509, 564, 605, 664, 714, 784, 839, 916, 980, 1070, 1141, 1239, 1320, 1435, 1522, 1644, 1745, 1886, 1993, 2143, 2264
Offset: 0

Views

Author

Ilya Gutkovskiy, Sep 09 2021

Keywords

Crossrefs

Cf. A005117, A073576, A308902, A347648, A347649, A347655, A347656.
Showing 1-10 of 10 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.