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Search: a308767 -id:a308767
Displaying 1-8 of 8 results found. page 1
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A341064 Number of ways to write n as an ordered sum of 4 squarefree numbers. +10
8
1, 4, 10, 16, 23, 32, 50, 68, 83, 92, 116, 148, 178, 192, 224, 276, 335, 360, 400, 460, 547, 580, 634, 704, 821, 868, 938, 1024, 1162, 1212, 1288, 1392, 1572, 1628, 1742, 1876, 2123, 2172, 2308, 2460, 2761, 2820, 2964, 3176, 3550, 3628, 3778, 4028, 4481, 4528, 4686, 4932, 5513, 5564 (list; graph; refs; listen; history; text; internal format)
OFFSET
4,2
LINKS
Table of n, a(n) for n=4..57.
FORMULA
G.f.: (Sum_{k>=1} mu(k)^2 * x^k)^4.
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, 4):
seq(a(n), n=4..57); # Alois P. Heinz, Feb 04 2021
MATHEMATICA
nmax = 57; CoefficientList[Series[Sum[MoebiusMu[k]^2 x^k, {k, 1, nmax}]^4, {x, 0, nmax}], x] // Drop[#, 4] &
CROSSREFS
Cf. A005117, A008683, A008966, A098235, A280210, A308767, A341065, A341066, A341067, A341068, A341069, A341070.
KEYWORD
nonn
AUTHOR
Ilya Gutkovskiy, Feb 04 2021
STATUS
approved
A341073 Number of partitions of n into 4 distinct squarefree parts. +10
8
1, 1, 1, 1, 2, 4, 3, 2, 5, 7, 8, 7, 11, 13, 15, 13, 17, 20, 23, 21, 28, 33, 34, 32, 40, 44, 47, 44, 55, 63, 66, 62, 75, 84, 87, 81, 98, 110, 115, 109, 127, 144, 148, 140, 159, 180, 186, 177, 199, 220, 231, 217, 241, 264, 275, 262, 290, 317, 325, 314, 343, 376, 382, 368, 403 (list; graph; refs; listen; history; text; internal format)
OFFSET
11,5
LINKS
Alois P. Heinz, Table of n, a(n) for n = 11..10000
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, 4):
seq(a(n), n=11..75); # Alois P. Heinz, Feb 04 2021
MATHEMATICA
b[n_, i_, t_] := b[n, i, t] = If[n == 0,
If[t == 0, 1, 0], If[i < 1 || t < 1, 0, b[n, i - 1, t] +
If[SquareFreeQ[i], b[n - i, Min[n - i, i - 1], t - 1], 0]]];
a[n_] := b[n, n, 4];
Table[a[n], {n, 11, 75}] (* Jean-François Alcover, Jul 14 2021, after Alois P. Heinz *)
CROSSREFS
Cf. A005117, A008966, A098236, A307835, A308767, A341064, A341074, A341075, A341095, A341096, A341097, A341098.
KEYWORD
nonn
AUTHOR
Ilya Gutkovskiy, Feb 04 2021
STATUS
approved
A308783 Sum of all the parts in the partitions of n into 4 squarefree parts. +10
5
0, 0, 0, 0, 4, 5, 12, 14, 32, 36, 60, 66, 96, 104, 154, 165, 240, 255, 342, 380, 500, 504, 660, 690, 888, 900, 1144, 1161, 1484, 1508, 1800, 1860, 2272, 2277, 2720, 2800, 3348, 3404, 4028, 4056, 4880, 4879, 5670, 5762, 6820, 6840, 7912, 8084, 9312, 9408 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,5
LINKS
Table of n, a(n) for n=0..49.
Index entries for sequences related to partitions
FORMULA
a(n) = n * Sum_{k=1..floor(n/4)} Sum_{j=k..floor((n-k)/3)} Sum_{i=j..floor((n-j-k)/2)} mu(k)^2 * mu(j)^2 * mu(i)^2 * mu(n-i-j-k)^2, where mu is the Möbius function (A008683).
a(n) = n * A308767(n).
a(n) = A308768(n) + A308762(n) + A308769(n) + A308770(n).
MATHEMATICA
Table[n*Sum[Sum[Sum[MoebiusMu[k]^2*MoebiusMu[j]^2*MoebiusMu[i]^2* MoebiusMu[n - i - j - k]^2, {i, j, Floor[(n - j - k)/2]}], {j, k, Floor[(n - k)/3]}], {k, Floor[n/4]}], {n, 0, 50}]
Table[Total[Flatten[Select[IntegerPartitions[n, {4}], AllTrue[#, SquareFreeQ]&]]], {n, 0, 50}] (* Harvey P. Dale, Aug 14 2022 *)
CROSSREFS
Cf. A008683, A308767, A308762, A308768, A308769, A308770.
KEYWORD
nonn
AUTHOR
Wesley Ivan Hurt, Jun 24 2019
STATUS
approved
A308762 Sum of the third largest parts of the partitions of n into 4 squarefree parts. +10
4
0, 0, 0, 0, 1, 1, 2, 3, 6, 6, 10, 11, 16, 16, 22, 23, 35, 38, 51, 57, 75, 76, 94, 99, 125, 128, 158, 162, 208, 209, 242, 251, 311, 317, 376, 390, 467, 478, 548, 553, 672, 682, 784, 801, 957, 957, 1096, 1101, 1284, 1294, 1471, 1469, 1725, 1717, 1917, 1918 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,7
LINKS
Table of n, a(n) for n=0..55.
Index entries for sequences related to partitions
FORMULA
a(n) = Sum_{k=1..floor(n/4)} Sum_{j=k..floor((n-k)/3)} Sum_{i=j..floor((n-j-k)/2)} mu(k)^2 * mu(j)^2 * mu(i)^2 * mu(n-i-j-k)^2 * j, where mu is the Möbius function (A008683).
a(n) = A308783(n) - A308768(n) - A308769(n) - A308770(n).
MATHEMATICA
Table[Sum[Sum[Sum[j * MoebiusMu[k]^2*MoebiusMu[j]^2*MoebiusMu[i]^2* MoebiusMu[n - i - j - k]^2, {i, j, Floor[(n - j - k)/2]}], {j, k, Floor[(n - k)/3]}], {k, Floor[n/4]}], {n, 0, 100}]
CROSSREFS
Cf. A008683, A308767, A308768, A308769, A308770, A308783.
KEYWORD
nonn
AUTHOR
Wesley Ivan Hurt, Jun 23 2019
STATUS
approved
A308768 Sum of the smallest parts of the partitions of n into 4 squarefree parts. +10
4
0, 0, 0, 0, 1, 1, 2, 2, 5, 5, 7, 8, 12, 11, 16, 16, 23, 23, 30, 32, 44, 43, 56, 57, 72, 72, 90, 87, 114, 112, 135, 137, 169, 164, 197, 196, 233, 238, 282, 276, 337, 332, 381, 378, 454, 447, 525, 523, 606, 609, 698, 678, 800, 799, 907, 895, 1050, 1022, 1157 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,7
LINKS
Table of n, a(n) for n=0..58.
Index entries for sequences related to partitions
FORMULA
a(n) = Sum_{k=1..floor(n/4)} Sum_{j=k..floor((n-k)/3)} Sum_{i=j..floor((n-j-k)/2)} mu(k)^2 * mu(j)^2 * mu(i)^2 * mu(n-i-j-k)^2 * k, where mu is the Möbius function (A008683).
a(n) = A308783(n) - A308762(n) - A308769(n) - A308770(n).
MATHEMATICA
Table[Sum[Sum[Sum[k * MoebiusMu[k]^2*MoebiusMu[j]^2*MoebiusMu[i]^2* MoebiusMu[n - i - j - k]^2, {i, j, Floor[(n - j - k)/2]}], {j, k, Floor[(n - k)/3]}], {k, Floor[n/4]}], {n, 0, 100}]
CROSSREFS
Cf. A008683, A308762, A308767, A308769, A308770, A308783.
KEYWORD
nonn
AUTHOR
Wesley Ivan Hurt, Jun 23 2019
STATUS
approved
A308769 Sum of the second largest parts of the partitions of n into 4 squarefree parts. +10
4
0, 0, 0, 0, 1, 1, 3, 4, 8, 8, 14, 15, 24, 25, 41, 45, 64, 64, 85, 93, 120, 123, 159, 172, 221, 222, 279, 291, 375, 386, 472, 494, 610, 612, 734, 745, 901, 899, 1075, 1067, 1297, 1272, 1493, 1490, 1765, 1757, 2046, 2076, 2398, 2408, 2743, 2774, 3187, 3177 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,7
LINKS
Table of n, a(n) for n=0..53.
Index entries for sequences related to partitions
FORMULA
a(n) = Sum_{k=1..floor(n/4)} Sum_{j=k..floor((n-k)/3)} Sum_{i=j..floor((n-j-k)/2)} mu(k)^2 * mu(j)^2 * mu(i)^2 * mu(n-i-j-k)^2 * i, where mu(n) is the Möbius function (A008683).
a(n) = A308783(n) - A308768(n) - A308762(n) - A308770(n).
MATHEMATICA
Table[Sum[Sum[Sum[i * MoebiusMu[k]^2*MoebiusMu[j]^2*MoebiusMu[i]^2* MoebiusMu[n - i - j - k]^2, {i, j, Floor[(n - j - k)/2]}], {j, k, Floor[(n - k)/3]}], {k, Floor[n/4]}], {n, 0, 100}]
CROSSREFS
Cf. A008683, A308762, A308767, A308768, A308770, A308783.
KEYWORD
nonn
AUTHOR
Wesley Ivan Hurt, Jun 23 2019
STATUS
approved
A308770 Sum of the largest parts of the partitions of n into 4 squarefree parts. +10
4
0, 0, 0, 0, 1, 2, 5, 5, 13, 17, 29, 32, 44, 52, 75, 81, 118, 130, 176, 198, 261, 262, 351, 362, 470, 478, 617, 621, 787, 801, 951, 978, 1182, 1184, 1413, 1469, 1747, 1789, 2123, 2160, 2574, 2593, 3012, 3093, 3644, 3679, 4245, 4384, 5024, 5097, 5738, 5891 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,6
LINKS
Table of n, a(n) for n=0..51.
Index entries for sequences related to partitions
FORMULA
a(n) = Sum_{k=1..floor(n/4)} Sum_{j=k..floor((n-k)/3)} Sum_{i=j..floor((n-j-k)/2)} mu(k)^2 * mu(j)^2 * mu(i)^2 * mu(n-i-j-k)^2 * (n-i-j-k) , where mu is the Möbius function (A008683).
a(n) = A308783(n) - A308768(n) - A308762(n) - A308769(n).
MATHEMATICA
Table[Sum[Sum[Sum[(n - i - j - k) * MoebiusMu[k]^2*MoebiusMu[j]^2* MoebiusMu[i]^2*MoebiusMu[n - i - j - k]^2, {i, j, Floor[(n - j - k)/2]}], {j, k, Floor[(n - k)/3]}], {k, Floor[n/4]}], {n, 0, 100}]
Table[Total[Select[IntegerPartitions[n, {4}], AllTrue[#, SquareFreeQ]&][[;; , 1]]], {n, 0, 60}] (* Harvey P. Dale, Oct 04 2023 *)
CROSSREFS
Cf. A008683, A308762, A308767, A308768, A308769, A308783.
KEYWORD
nonn
AUTHOR
Wesley Ivan Hurt, Jun 23 2019
STATUS
approved
A347655 Number of partitions of n into at most 4 squarefree parts. +10
3
1, 1, 2, 3, 4, 5, 7, 8, 10, 11, 13, 14, 17, 19, 22, 24, 28, 31, 34, 37, 42, 44, 50, 53, 59, 61, 69, 71, 80, 82, 90, 93, 103, 106, 117, 121, 134, 137, 150, 154, 169, 173, 188, 194, 212, 216, 235, 240, 259, 264, 284, 288, 310, 314, 337, 342, 368, 370, 398, 403, 432 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
LINKS
Table of n, a(n) for n=0..60.
MATHEMATICA
Table[Count[IntegerPartitions[n, 4], _?(AllTrue[#, SquareFreeQ]&)], {n, 0, 100}] (* Harvey P. Dale, Sep 04 2023 *)
CROSSREFS
Cf. A005117, A073576, A308767, A347648, A347649, A347656, A347657.
KEYWORD
nonn
AUTHOR
Ilya Gutkovskiy, Sep 09 2021
STATUS
approved
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Last modified September 11 10:08 EDT 2024. Contains 375827 sequences. (Running on oeis4.)