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Search: a308907 -id:a308907
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Number of partitions of n into 6 squarefree parts.
+10
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
OFFSET
0,9
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}]
KEYWORD
nonn
AUTHOR
Wesley Ivan Hurt, Jun 29 2019
STATUS
approved
Sum of all the parts in the partitions of n into 6 squarefree parts.
+10
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
OFFSET
0,7
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}]
KEYWORD
nonn
AUTHOR
Wesley Ivan Hurt, Jun 29 2019
STATUS
approved
Sum of the smallest parts in the partitions of n into 6 squarefree parts.
+10
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
OFFSET
0,9
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}]
KEYWORD
nonn
AUTHOR
Wesley Ivan Hurt, Jun 29 2019
STATUS
approved
Sum of the fourth largest parts in the partitions of n into 6 squarefree parts.
+10
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
OFFSET
0,9
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}]
KEYWORD
nonn
AUTHOR
Wesley Ivan Hurt, Jun 29 2019
STATUS
approved
Sum of the third largest parts in the partitions of n into 6 squarefree parts.
+10
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
OFFSET
0,9
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 *)
KEYWORD
nonn
AUTHOR
Wesley Ivan Hurt, Jun 29 2019
STATUS
approved
Sum of the second largest parts in the partitions of n into 6 squarefree parts.
+10
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
OFFSET
0,9
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 *)
KEYWORD
nonn
AUTHOR
Wesley Ivan Hurt, Jun 29 2019
STATUS
approved
Sum of the largest parts in the partitions of n into 6 squarefree parts.
+10
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
OFFSET
0,8
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}]
KEYWORD
nonn
AUTHOR
Wesley Ivan Hurt, Jun 29 2019
STATUS
approved

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