cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-10 of 17 results. Next

A300663 Expansion of 1/(1 - Sum_{k>=1} mu(k)*x^k), where mu() is the Moebius function (A008683).

Original entry on oeis.org

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
Offset: 0

Views

Author

Ilya Gutkovskiy, Mar 10 2018

Keywords

Comments

Invert transform of A008683.

Links

Crossrefs

Cf. A008683, A050385, A068341, A073776, A073777, A117209, A117210, A185694, A280194.

Programs

  • 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}]
  • 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

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

A331846 Number of compositions (ordered partitions) of n into distinct squarefree parts.

Original entry on oeis.org

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
Offset: 0

Views

Author

Ilya Gutkovskiy, Jan 29 2020

Keywords

Examples

			a(7) = 5 because we have [7], [6, 1], [5, 2], [2, 5] and [1, 6].

Links

Crossrefs

Cf. A005117, A032020, A032021, A032022, A087188, A218396, A219107, A280194, A331843, A331844, A331845, A331847.

A347777 Number of compositions (ordered partitions) of n into at most 2 squarefree parts.

Original entry on oeis.org

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
Offset: 0

Views

Author

Ilya Gutkovskiy, Sep 13 2021

Keywords

Crossrefs

Cf. A005117, A098235, A280194, A347778, A347779, A347780.

Programs

  • Mathematica
    Table[Length@Flatten[Permutations/@IntegerPartitions[n,2,Select[Range@n,SquareFreeQ]],1],{n,0,100}] (* Giorgos Kalogeropoulos, Sep 13 2021 *)

A347778 Number of compositions (ordered partitions) of n into at most 3 squarefree parts.

Original entry on oeis.org

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
Offset: 0

Views

Author

Ilya Gutkovskiy, Sep 13 2021

Keywords

Crossrefs

Cf. A005117, A280194, A280210, A347777, A347779, A347780.

Programs

  • Mathematica
    Table[Length@Flatten[Permutations/@IntegerPartitions[n,3,Select[Range@n,SquareFreeQ]],1],{n,0,58}] (* Giorgos Kalogeropoulos, Sep 13 2021 *)

A347779 Number of compositions (ordered partitions) of n into at most 4 squarefree parts.

Original entry on oeis.org

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
Offset: 0

Views

Author

Ilya Gutkovskiy, Sep 13 2021

Keywords

Crossrefs

Cf. A005117, A280194, A341064, A347777, A347778, A347780.

Programs

  • Mathematica
    Table[Length@Flatten[Permutations/@IntegerPartitions[n,4,Select[Range@n,SquareFreeQ]],1],{n,0,51}] (* Giorgos Kalogeropoulos, Sep 13 2021 *)

A284464 Number of compositions (ordered partitions) of n into squarefree divisors of n.

Original entry on oeis.org

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
Offset: 0

Views

Author

Ilya Gutkovskiy, Mar 27 2017

Keywords

Examples

			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].

Links

Crossrefs

Cf. A005117, A008683, A100346, A225244, A225245, A280194.

Programs

  • 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}]
  • 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

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.

A300706 Number of compositions (ordered partitions) of n into squarefree parts that do not divide n.

Original entry on oeis.org

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
Offset: 0

Views

Author

Ilya Gutkovskiy, Mar 11 2018

Keywords

Examples

			a(18) = 4 because we have [13, 5], [11, 7], [7, 11] and [5, 13].

Links

Crossrefs

Cf. A005117, A280194, A284464, A300585, A300586, A300702, A300703, A300704.

Programs

  • 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}]

A347780 Number of compositions (ordered partitions) of n into at most 5 squarefree parts.

Original entry on oeis.org

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
Offset: 0

Views

Author

Ilya Gutkovskiy, Sep 13 2021

Keywords

Crossrefs

Cf. A005117, A280194, A341065, A347777, A347778, A347779.

Programs

  • Mathematica
    Table[Length@Flatten[Permutations/@IntegerPartitions[n,5,Select[Range@n,SquareFreeQ]],1],{n,0,45}] (* Giorgos Kalogeropoulos, Sep 13 2021 *)

A280197 Expansion of 1/(1 - Sum_{k>=2} mu(k)^2*x^k), where mu(k) is the Moebius function (A008683).

Original entry on oeis.org

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
Offset: 0

Views

Author

Ilya Gutkovskiy, Dec 28 2016

Keywords

Comments

Number of compositions (ordered partitions) into squarefree parts > 1 (A144338).

Examples

			a(5) = 3 because we have [5], [3, 2] and [2, 3].

Links

Crossrefs

Cf. A005117, A008683, A073576, A144338, A280127, A280194.

Programs

  • 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]

Formula

G.f.: 1/(1 - Sum_{k>=2} mu(k)^2*x^k).

A280198 Expansion of 1/(1 - Sum_{k>=1} mu(2*k-1)^2*x^(2*k-1)), where mu() is the Moebius function (A008683).

Original entry on oeis.org

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
Offset: 0

Views

Author

Ilya Gutkovskiy, Dec 28 2016

Keywords

Comments

Number of compositions (ordered partitions) into odd squarefree parts (A056911).

Examples

			a(4) = 3 because we have [3, 1], [1, 3] and [1, 1, 1, 1].

Links

Crossrefs

Cf. A005117, A008683, A056911, A134345, A280194.

Programs

  • Mathematica
    nmax = 46; CoefficientList[Series[1/(1 - Sum[MoebiusMu[2 k - 1]^2 x^(2 k - 1), {k, 1, nmax}]), {x, 0, nmax}], x]

Formula

G.f.: 1/(1 - Sum_{k>=1} mu(2*k-1)^2*x^(2*k-1)).
Showing 1-10 of 17 results. Next

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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