A308907 -id:A308907

A308902

Number of partitions of n into 6 squarefree parts.

+0
10

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,9

LINKS

Table of n, a(n) for n=0..58.

Index entries for sequences related to partitions

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.

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}]

CROSSREFS

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

KEYWORD

nonn

AUTHOR

Wesley Ivan Hurt, Jun 29 2019

STATUS

approved

A308903

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

+0
7

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,7

LINKS

Table of n, a(n) for n=0..47.

Index entries for sequences related to partitions

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), 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).

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}]

CROSSREFS

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

KEYWORD

nonn

AUTHOR

Wesley Ivan Hurt, Jun 29 2019

STATUS

approved

A308906

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

+0
7

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,9

LINKS

Table of n, a(n) for n=0..56.

Index entries for sequences related to partitions

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).

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}]

CROSSREFS

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

KEYWORD

nonn

AUTHOR

Wesley Ivan Hurt, Jun 29 2019

STATUS

approved

A308908

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

+0
7

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,9

LINKS

Table of n, a(n) for n=0..54.

Index entries for sequences related to partitions

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).

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}]

CROSSREFS

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

KEYWORD

nonn

AUTHOR

Wesley Ivan Hurt, Jun 29 2019

STATUS

approved

A308909

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

+0
7

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,9

LINKS

Table of n, a(n) for n=0..53.

Index entries for sequences related to partitions

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).

MATHEMATICA

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

CROSSREFS

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

KEYWORD

nonn

AUTHOR

Wesley Ivan Hurt, Jun 29 2019

STATUS

approved

A308910

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

+0
7

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,9

LINKS

Table of n, a(n) for n=0..51.

Index entries for sequences related to partitions

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).

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 *)

CROSSREFS

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

KEYWORD

nonn

AUTHOR

Wesley Ivan Hurt, Jun 29 2019

STATUS

approved

A308911

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

+0
7

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,8

LINKS

Table of n, a(n) for n=0..50.

Index entries for sequences related to partitions

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).

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}]

CROSSREFS

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

KEYWORD

nonn

AUTHOR

Wesley Ivan Hurt, Jun 29 2019

STATUS

approved