Search: a308767 -id:a308767
|
|
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
|
|
|
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):
|
|
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
|
|
|
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
|
|
|
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):
|
|
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];
|
|
CROSSREFS
|
Cf. A005117, A008966, A098236, A307835, A308767, A341064, A341074, A341075, A341095, A341096, A341097, A341098.
|
|
KEYWORD
|
nonn
|
|
AUTHOR
|
|
|
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
|
|
|
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).
|
|
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
|
|
|
KEYWORD
|
nonn
|
|
AUTHOR
|
|
|
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
|
|
|
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).
|
|
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
|
|
|
KEYWORD
|
nonn
|
|
AUTHOR
|
|
|
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
|
|
|
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).
|
|
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
|
|
|
KEYWORD
|
nonn
|
|
AUTHOR
|
|
|
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
|
|
|
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).
|
|
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
|
|
|
KEYWORD
|
nonn
|
|
AUTHOR
|
|
|
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
|
|
|
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).
|
|
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
|
|
|
KEYWORD
|
nonn
|
|
AUTHOR
|
|
|
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
|
|
|
MATHEMATICA
|
Table[Count[IntegerPartitions[n, 4], _?(AllTrue[#, SquareFreeQ]&)], {n, 0, 100}] (* Harvey P. Dale, Sep 04 2023 *)
|
|
CROSSREFS
|
|
|
KEYWORD
|
nonn
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
Search completed in 0.007 seconds
|