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Search: a280194 -id:a280194
Displaying 1-10 of 17 results found. page 1 2
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A300663 Expansion of 1/(1 - Sum_{k>=1} mu(k)*x^k), where mu() is the Moebius function (A008683). +10
9
1, 1, 0, -2, -3, -2, 3, 8, 8, -2, -16, -24, -10, 24, 59, 54, -11, -117, -174, -90, 162, 431, 449, -20, -835, -1393, -848, 1062, 3352, 3748, 317, -6257, -11134, -7583, 7294, 25956, 30786, 5217, -46545, -88132, -65062, 48534, 199234, 249263, 63034, -342174, -691679, -554002 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,4
COMMENTS
Invert transform of A008683.
LINKS
Seiichi Manyama, Table of n, a(n) for n = 0..5000
N. J. A. Sloane, Transforms
FORMULA
G.f.: 1/(1 - Sum_{k>=1} A008683(k)*x^k).
a(0) = 1; a(n) = Sum_{k=1..n} mu(k) * a(n-k). - Seiichi Manyama, Apr 06 2022
MAPLE
a:= proc(n) option remember; `if`(n=0, 1, add(
numtheory[mobius](j)*a(n-j), j=1..n))
end:
seq(a(n), n=0..50); # Alois P. Heinz, Mar 10 2018
MATHEMATICA
nmax = 47; CoefficientList[Series[1/(1 - Sum[MoebiusMu[k] x^k, {k, 1, nmax}]), {x, 0, nmax}], x]
a[0] = 1; a[n_] := a[n] = Sum[MoebiusMu[k] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 47}]
PROG
(PARI) my(N=66, x='x+O('x^N)); Vec(1/(1-sum(k=1, N, moebius(k)*x^k))) \\ Seiichi Manyama, Apr 06 2022
(PARI) a(n) = if(n==0, 1, sum(k=1, n, moebius(k)*a(n-k))); \\ Seiichi Manyama, Apr 06 2022
CROSSREFS
Cf. A008683, A050385, A068341, A073776, A073777, A117209, A117210, A185694, A280194.
KEYWORD
sign
AUTHOR
Ilya Gutkovskiy, Mar 10 2018
STATUS
approved
A331846 Number of compositions (ordered partitions) of n into distinct squarefree parts. +10
5
1, 1, 1, 3, 2, 3, 9, 5, 12, 16, 21, 41, 42, 49, 59, 79, 130, 231, 230, 295, 226, 495, 609, 699, 1472, 1042, 1377, 2308, 2982, 3425, 3879, 4877, 7156, 7189, 13531, 14797, 13570, 19551, 27667, 30327, 36382, 47519, 60783, 70561, 78330, 136988, 121659, 174851 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,4
LINKS
Table of n, a(n) for n=0..47.
Index entries for sequences related to compositions
EXAMPLE
a(7) = 5 because we have [7], [6, 1], [5, 2], [2, 5] and [1, 6].
CROSSREFS
Cf. A005117, A032020, A032021, A032022, A087188, A218396, A219107, A280194, A331843, A331844, A331845, A331847.
KEYWORD
nonn
AUTHOR
Ilya Gutkovskiy, Jan 29 2020
STATUS
approved
A347777 Number of compositions (ordered partitions) of n into at most 2 squarefree parts. +10
5
1, 1, 2, 3, 3, 3, 4, 5, 6, 4, 4, 5, 7, 7, 6, 7, 10, 9, 8, 7, 11, 9, 10, 9, 14, 10, 10, 10, 13, 11, 10, 11, 16, 13, 14, 13, 22, 15, 14, 15, 22, 17, 16, 19, 25, 20, 16, 17, 26, 20, 16, 15, 27, 21, 20, 15, 26, 21, 22, 19, 29, 23, 22, 22, 30, 23, 22, 23, 35, 25, 26 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
LINKS
Table of n, a(n) for n=0..70.
MATHEMATICA
Table[Length@Flatten[Permutations/@IntegerPartitions[n, 2, Select[Range@n, SquareFreeQ]], 1], {n, 0, 100}] (* Giorgos Kalogeropoulos, Sep 13 2021 *)
CROSSREFS
Cf. A005117, A098235, A280194, A347778, A347779, A347780.
KEYWORD
nonn
AUTHOR
Ilya Gutkovskiy, Sep 13 2021
STATUS
approved
A347778 Number of compositions (ordered partitions) of n into at most 3 squarefree parts. +10
5
1, 1, 2, 4, 6, 9, 11, 14, 18, 23, 25, 26, 28, 37, 42, 44, 46, 57, 66, 70, 68, 79, 88, 96, 92, 106, 115, 124, 118, 134, 143, 149, 142, 161, 176, 187, 178, 210, 221, 235, 214, 251, 266, 280, 262, 300, 328, 335, 308, 350, 379, 385, 342, 396, 425, 447, 392, 442, 475 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
LINKS
Table of n, a(n) for n=0..58.
MATHEMATICA
Table[Length@Flatten[Permutations/@IntegerPartitions[n, 3, Select[Range@n, SquareFreeQ]], 1], {n, 0, 58}] (* Giorgos Kalogeropoulos, Sep 13 2021 *)
CROSSREFS
Cf. A005117, A280194, A280210, A347777, A347779, A347780.
KEYWORD
nonn
AUTHOR
Ilya Gutkovskiy, Sep 13 2021
STATUS
approved
A347779 Number of compositions (ordered partitions) of n into at most 4 squarefree parts. +10
5
1, 1, 2, 4, 7, 13, 21, 30, 41, 55, 75, 94, 111, 129, 158, 192, 224, 249, 290, 346, 403, 439, 488, 556, 639, 686, 749, 828, 939, 1002, 1081, 1173, 1304, 1373, 1464, 1579, 1750, 1838, 1963, 2111, 2337, 2423, 2574, 2740, 3023, 3120, 3292, 3511, 3858, 3978, 4157, 4413 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
LINKS
Table of n, a(n) for n=0..51.
MATHEMATICA
Table[Length@Flatten[Permutations/@IntegerPartitions[n, 4, Select[Range@n, SquareFreeQ]], 1], {n, 0, 51}] (* Giorgos Kalogeropoulos, Sep 13 2021 *)
CROSSREFS
Cf. A005117, A280194, A341064, A347777, A347778, A347780.
KEYWORD
nonn
AUTHOR
Ilya Gutkovskiy, Sep 13 2021
STATUS
approved
A300706 Number of compositions (ordered partitions) of n into squarefree parts that do not divide n. +10
4
1, 0, 0, 0, 0, 2, 0, 5, 2, 5, 2, 27, 2, 67, 12, 16, 28, 366, 4, 848, 28, 182, 153, 4591, 20, 4172, 554, 2217, 558, 57695, 6, 134118, 3834, 14629, 6972, 97478, 258, 1684852, 24467, 120869, 5308, 9104710, 189, 21165023, 124427, 117017, 297830, 114373157, 3394, 126979537, 72158, 7655405 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,6
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..2000
Index entries for sequences related to compositions
EXAMPLE
a(18) = 4 because we have [13, 5], [11, 7], [7, 11] and [5, 13].
MAPLE
with(numtheory):
a:= proc(m) option remember; local b; b:= proc(n) option
remember; `if`(n=0, 1, add(`if`(not issqrfree(j) or
irem(m, j)=0, 0, b(n-j)), j=2..n)) end; b(m)
end:
seq(a(n), n=0..70); # Alois P. Heinz, Mar 11 2018
MATHEMATICA
Table[SeriesCoefficient[1/(1 - Sum[Boole[Mod[n, k] != 0 && SquareFreeQ[k]] x^k, {k, 1, n}]), {x, 0, n}], {n, 0, 51}]
CROSSREFS
Cf. A005117, A280194, A284464, A300585, A300586, A300702, A300703, A300704.
KEYWORD
nonn
AUTHOR
Ilya Gutkovskiy, Mar 11 2018
STATUS
approved
A347780 Number of compositions (ordered partitions) of n into at most 5 squarefree parts. +10
4
1, 1, 2, 4, 7, 14, 26, 45, 71, 105, 151, 214, 291, 379, 473, 593, 744, 919, 1095, 1301, 1563, 1884, 2203, 2536, 2929, 3427, 3929, 4433, 4979, 5692, 6422, 7158, 7904, 8863, 9844, 10830, 11810, 13078, 14378, 15706, 17007, 18718, 20424, 22165, 23803, 26025 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
LINKS
Table of n, a(n) for n=0..45.
MATHEMATICA
Table[Length@Flatten[Permutations/@IntegerPartitions[n, 5, Select[Range@n, SquareFreeQ]], 1], {n, 0, 45}] (* Giorgos Kalogeropoulos, Sep 13 2021 *)
CROSSREFS
Cf. A005117, A280194, A341065, A347777, A347778, A347779.
KEYWORD
nonn
AUTHOR
Ilya Gutkovskiy, Sep 13 2021
STATUS
approved
A284464 Number of compositions (ordered partitions) of n into squarefree divisors of n. +10
3
1, 1, 2, 2, 5, 2, 25, 2, 34, 19, 129, 2, 1046, 2, 742, 450, 1597, 2, 44254, 2, 27517, 3321, 29967, 2, 1872757, 571, 200390, 18560, 854850, 2, 154004511, 2, 3524578, 226020, 9262157, 51886, 3353855285, 2, 63346598, 2044895, 1255304727, 2, 185493291001, 2, 1282451595, 345852035, 2972038875, 2, 6006303471178 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..2000
Eric Weisstein's World of Mathematics, Squarefree
Index entries for sequences related to compositions
FORMULA
a(n) = [x^n] 1/(1 - Sum_{d|n, |mu(d)| = 1} x^d), where mu(d) is the Moebius function (A008683).
a(n) = 2 if n is a prime.
EXAMPLE
a(4) = 5 because 4 has 3 divisors {1, 2, 4} among which 2 are squarefree {1, 2} therefore we have [2, 2], [2, 1, 1], [1, 2, 1], [1, 2, 2] and [1, 1, 1, 1].
MAPLE
with(numtheory):
a:= proc(n) option remember; local b, l;
l, b:= select(issqrfree, divisors(n)),
proc(m) option remember; `if`(m=0, 1,
add(`if`(j>m, 0, b(m-j)), j=l))
end; b(n)
end:
seq(a(n), n=0..50); # Alois P. Heinz, Mar 30 2017
MATHEMATICA
Table[d = Divisors[n]; Coefficient[Series[1/(1 - Sum[MoebiusMu[d[[k]]]^2 x^d[[k]], {k, Length[d]}]), {x, 0, n}], x, n], {n, 0, 48}]
PROG
(Python)
from sympy import divisors
from sympy.ntheory.factor_ import core
from sympy.core.cache import cacheit
@cacheit
def a(n):
l=[x for x in divisors(n) if core(x)==x]
@cacheit
def b(m): return 1 if m==0 else sum(b(m - j) for j in l if j <= m)
return b(n)
print([a(n) for n in range(51)]) # Indranil Ghosh, Aug 01 2017, after Maple code
CROSSREFS
Cf. A005117, A008683, A100346, A225244, A225245, A280194.
KEYWORD
nonn
AUTHOR
Ilya Gutkovskiy, Mar 27 2017
STATUS
approved
A280197 Expansion of 1/(1 - Sum_{k>=2} mu(k)^2*x^k), where mu(k) is the Moebius function (A008683). +10
1
1, 0, 1, 1, 1, 3, 3, 6, 8, 12, 20, 28, 45, 68, 102, 159, 238, 367, 557, 849, 1298, 1973, 3015, 4592, 7002, 10679, 16276, 24822, 37841, 57696, 87971, 134119, 204497, 311783, 475370, 724786, 1105053, 1684853, 2568837, 3916642, 5971587, 9104711, 13881698, 21165024, 32269721, 49200718, 75014949, 114373158, 174381511 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,6
COMMENTS
Number of compositions (ordered partitions) into squarefree parts > 1 (A144338).
LINKS
Robert Israel, Table of n, a(n) for n = 0..5456
Eric Weisstein's World of Mathematics, Squarefree
Index entries for sequences related to compositions
FORMULA
G.f.: 1/(1 - Sum_{k>=2} mu(k)^2*x^k).
EXAMPLE
a(5) = 3 because we have [5], [3, 2] and [2, 3].
MAPLE
N:= 100: # for a(0)..a(N)
g:= 1/(1-add(numtheory:-mobius(k)^2*x^k, k=2..N)):
S:= series(g, x, N+1):
seq(coeff(S, x, j), j=0..N); # Robert Israel, Dec 29 2016
MATHEMATICA
nmax = 48; CoefficientList[Series[1/(1 - Sum[MoebiusMu[k]^2 x^k, {k, 2, nmax}]), {x, 0, nmax}], x]
CROSSREFS
Cf. A005117, A008683, A073576, A144338, A280127, A280194.
KEYWORD
nonn
AUTHOR
Ilya Gutkovskiy, Dec 28 2016
STATUS
approved
A280198 Expansion of 1/(1 - Sum_{k>=1} mu(2*k-1)^2*x^(2*k-1)), where mu() is the Moebius function (A008683). +10
1
1, 1, 1, 2, 3, 5, 8, 13, 21, 33, 53, 86, 138, 222, 357, 574, 923, 1484, 2387, 3839, 6173, 9927, 15964, 25672, 41284, 66389, 106762, 171686, 276091, 443989, 713988, 1148179, 1846411, 2969252, 4774918, 7678647, 12348195, 19857396, 31933099, 51352294, 82580715, 132799801, 213558181, 343427445, 552272966, 888121883, 1428207656 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,4
COMMENTS
Number of compositions (ordered partitions) into odd squarefree parts (A056911).
LINKS
Table of n, a(n) for n=0..46.
Eric Weisstein's World of Mathematics, Squarefree
Index entries for sequences related to compositions
FORMULA
G.f.: 1/(1 - Sum_{k>=1} mu(2*k-1)^2*x^(2*k-1)).
EXAMPLE
a(4) = 3 because we have [3, 1], [1, 3] and [1, 1, 1, 1].
MATHEMATICA
nmax = 46; CoefficientList[Series[1/(1 - Sum[MoebiusMu[2 k - 1]^2 x^(2 k - 1), {k, 1, nmax}]), {x, 0, nmax}], x]
CROSSREFS
Cf. A005117, A008683, A056911, A134345, A280194.
KEYWORD
nonn
AUTHOR
Ilya Gutkovskiy, Dec 28 2016
STATUS
approved
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Last modified August 29 18:55 EDT 2024. Contains 375518 sequences. (Running on oeis4.)