A087188 -id:A087188 - cp's OEIS frontend

A292561 Expansion of Product_{k>=1} (1 - mu(k)^2*x^k), where mu() is the Moebius function (A008683).

1, -1, -1, 0, 1, 0, -1, 1, 2, 0, -3, 0, 2, 0, -3, 0, 5, 0, -4, -2, 4, 0, -5, 0, 7, 3, -8, -1, 5, 1, -10, 0, 13, 2, -10, -3, 14, -2, -17, -3, 21, 5, -22, 0, 22, 4, -34, -5, 33, 9, -33, -10, 43, 6, -43, -19, 52, 16, -51, -13, 56, 24, -71, -20, 64, 26, -78, -24, 90, 24, -90, -39, 112, 26, -115, -37

Offset: 0

Ilya Gutkovskiy, Sep 19 2017

Convolution inverse of A073576.

The difference between the number of partitions of n into an even number of distinct squarefree parts and the number of partitions of n into an odd number of distinct squarefree parts.

Alois P. Heinz, Table of n, a(n) for n = 0..10000
Index entries for related partition-counting sequences

Cf. A005117, A008683, A046675, A073576, A087188.

with(numtheory):
b:= proc(n) option remember; `if`(n=0, 1, add(add(d*
      abs(mobius(d)), d=divisors(j)) *b(n-j), j=1..n)/n)
    end:
a:= proc(n) option remember; `if`(n=0, 1,
      -add(b(n-i)*a(i), i=0..n-1))
    end:
seq(a(n), n=0..80);  # Alois P. Heinz, Sep 20 2017

nmax = 75; CoefficientList[Series[Product[1 - MoebiusMu[k]^2 x^k, {k, 1, nmax}], {x, 0, nmax}], x]

G.f.: Product_{k>=1} (1 - x^A005117(k)).

A280128 Expansion of Product_{k>=2} (1 + mu(k)^2*x^k), where mu(k) is the Moebius function (A008683).

Original entry on oeis.org

1, 0, 1, 1, 0, 2, 1, 2, 2, 2, 3, 3, 3, 5, 4, 6, 7, 7, 9, 9, 11, 13, 14, 16, 19, 18, 24, 23, 28, 31, 33, 39, 42, 46, 52, 57, 63, 71, 76, 87, 92, 103, 113, 123, 135, 149, 161, 178, 193, 210, 231, 249, 274, 298, 323, 352, 382, 414, 451, 486, 528, 572, 617, 669

Offset: 0

Ilya Gutkovskiy, Dec 26 2016

nonn

Number of partitions of n into distinct squarefree parts > 1 (A144338).

			G.f. = 1 + x^2 + x^3 + 2*x^5 + x^6 + 2*x^7 + 2*x^8 + 2*x^9 + 3*x^10 + 3*x^11 + ...
a(10) = 3 because we have [10], [7, 3] and [5, 3, 2].

G. C. Greubel, Table of n, a(n) for n = 0..5000
Joerg Arndt, Matters Computational (The Fxtbook), section 16.4.3 "Partitions into square-free parts", pp.351-352
Eric Weisstein's World of Mathematics, Squarefree
Index entries for related partition-counting sequences

Cf. A005117, A008683, A087188, A144338, A280127.

with(numtheory): seq(coeff(series(mul(1+mobius(k)^2*x^k,k=2..n), x,n+1),x,n),n=0..70); # Muniru A Asiru, Jul 30 2018

nmax = 75; CoefficientList[Series[Product[1 + MoebiusMu[k]^2 x^k, {k, 2, nmax}], {x, 0, nmax}], x]

{a(n) = if(n < 0, 0, polcoeff( prod(k=2, n, 1 + issquarefree(k)*x^k + x*O(x^n)), n))}; /* Michael Somos, Dec 26 2016 */

G.f.: Product_{k>=2} (1 + mu(k)^2*x^k).

A300894 L.g.f.: log(Product_{k>=1} (1 + mu(k)^2x^k)) = Sum_{n>=1} a(n)x^n/n, where mu() is the Moebius function (A008683).

Original entry on oeis.org

1, 1, 4, -3, 6, 4, 8, -3, 4, 6, 12, -12, 14, 8, 24, -3, 18, 4, 20, -18, 32, 12, 24, -12, 6, 14, 4, -24, 30, 24, 32, -3, 48, 18, 48, -12, 38, 20, 56, -18, 42, 32, 44, -36, 24, 24, 48, -12, 8, 6, 72, -42, 54, 4, 72, -24, 80, 30, 60, -72, 62, 32, 32, -3, 84, 48, 68, -54, 96, 48, 72, -12, 74, 38, 24

Offset: 1

Ilya Gutkovskiy, Mar 14 2018

			L.g.f.: L(x) = x + x^2/2 + 4*x^3/3 - 3*x^4/4 + 6*x^5/5 + 4*x^6/6 + 8*x^7/7 - 3*x^8/8 + 4*x^9/9 + 6*x^10/10 + ...
exp(L(x)) = 1 + x + x^2 + 2*x^3 + x^4 + 2*x^5 + 3*x^6 + 3*x^7 + 4*x^8 + 4*x^9 + 5*x^10 + ... + A087188(n)*x^n + ...

Andrew Howroyd, Table of n, a(n) for n = 1..1000

Cf. A005117, A008683, A048250, A065091, A087188, A300852.

nmax = 75; Rest[CoefficientList[Series[Log[Product[(1 + MoebiusMu[k]^2 x^k), {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]]
nmax = 75; Rest[CoefficientList[Series[Sum[MoebiusMu[k]^2 k x^k/(1 + x^k), {k, 1, nmax}], {x, 0, nmax}], x]]
Table[DivisorSum[n, (-1)^(n/# + 1) # &, SquareFreeQ[#] &], {n, 75}]
f[p_, e_] := If[p == 2, If[e == 1, 1, -3], p + 1]; a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Oct 25 2020 *)

seq(n)=Vec(sum(k=1, n, moebius(k)^2*k*x^k/(1 + x^k) + O(x*x^n))); \\ Andrew Howroyd, Jul 20 2018

a(n)={sumdiv(n, d, if(issquarefree(d), (-1)^(n/d + 1) * d))} \\ Andrew Howroyd, Jul 20 2018

G.f.: Sum_{k>=1} mu(k)^2*k*x^k/(1 + x^k).
a(n) = n + 1 if n is an odd prime (A065091).
Multiplicative with a(2^e) = 1 if e = 1, and -3 otherwise, and a(p^e) = p+1 for an odd prime p. - Amiram Eldar, Oct 25 2020
Sum_{k=1..n} a(k) ~ n^2/4. - Amiram Eldar, Nov 20 2022

Keyword:mult added by Andrew Howroyd, Jul 20 2018

A300586 Number of partitions of n into distinct squarefree parts that do not divide n.

Original entry on oeis.org

1, 0, 0, 0, 0, 1, 0, 1, 1, 1, 1, 2, 1, 4, 2, 2, 4, 6, 2, 8, 4, 6, 6, 15, 4, 11, 10, 12, 8, 30, 3, 38, 24, 17, 24, 23, 14, 70, 36, 37, 23, 102, 8, 122, 49, 39, 80, 177, 38, 136, 54, 113, 101, 297, 60, 152, 102, 192, 226, 485, 28, 571, 312, 200, 390, 338, 84, 908, 393, 507, 104, 1229, 241, 1421

Offset: 0

Ilya Gutkovskiy, Mar 09 2018

nonn

			a(14) = 2 because we have [11, 3] and [6, 5, 3].

Index entries for sequences related to partitions

Cf. A005117, A087188, A200745, A209402, A225245, A300584, A300585.

Table[SeriesCoefficient[Product[(1 + Boole[Mod[n, k] != 0 && SquareFreeQ[k]] x^k), {k, 1, n}], {x, 0, n}], {n, 0, 73}]

A357525 Expansion of Product_{k>=1} (1 + mu(k)*x^k).

Original entry on oeis.org

1, 1, -1, -2, -1, 0, 1, 1, 0, 0, 1, 0, -2, -2, 1, 4, 3, -2, -4, -2, 0, 2, 3, 0, -1, 1, 0, -3, -3, -1, 2, 4, 3, 0, -2, -1, 2, 0, -5, -3, 3, 3, 0, -2, -4, -2, 4, 5, 3, 3, 1, -4, -9, -8, 3, 11, 6, 0, -3, -7, -4, 2, -1, -2, 6, 8, -2, -10, -8, 4, 14, 11, 2, -6, -11, -5

Offset: 0

Ilya Gutkovskiy, Oct 02 2022

sign

Cf. A008683, A087188, A117210, A185694, A300663, A306327, A357521, A357524.

nmax = 75; CoefficientList[Series[Product[(1 + MoebiusMu[k] x^k), {k, 1, nmax}], {x, 0, nmax}], x]
a[0] = 1; a[n_] := a[n] = -(1/n) Sum[Sum[d (-MoebiusMu[d])^(k/d), {d, Divisors[k]}] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 75}]

A358023 Number of partitions of n into at most 2 distinct squarefree parts.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 2, 3, 3, 2, 2, 3, 3, 4, 3, 4, 5, 5, 4, 4, 5, 5, 5, 5, 7, 5, 5, 5, 6, 6, 5, 6, 8, 7, 7, 7, 11, 8, 7, 8, 11, 9, 8, 10, 12, 10, 8, 9, 13, 10, 8, 8, 13, 11, 10, 8, 13, 11, 11, 10, 14, 12, 11, 11, 15, 12, 11, 12, 17, 13, 13, 12, 21, 14, 14, 13, 19, 15, 13, 15, 20

Offset: 0

Ilya Gutkovskiy, Oct 25 2022

nonn

Cf. A005117, A087188, A098236, A347648, A347777, A358024, A358025.

A358024 Number of partitions of n into at most 3 distinct squarefree parts.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 3, 3, 4, 4, 5, 5, 5, 7, 8, 8, 9, 10, 13, 12, 13, 14, 17, 16, 18, 17, 21, 20, 21, 23, 26, 25, 26, 27, 32, 31, 33, 36, 40, 40, 39, 42, 48, 47, 47, 50, 58, 56, 55, 58, 66, 64, 61, 67, 75, 74, 70, 74, 84, 83, 79, 82, 93, 91, 89, 93, 103, 102, 97, 105, 115

Offset: 0

Ilya Gutkovskiy, Oct 25 2022

nonn

Cf. A005117, A087188, A307835, A347649, A347778, A358023, A358025.

A358025 Number of partitions of n into at most 4 distinct squarefree parts.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 3, 3, 4, 4, 5, 6, 6, 8, 9, 10, 13, 13, 15, 17, 20, 22, 24, 27, 31, 32, 34, 37, 41, 46, 47, 53, 59, 61, 64, 71, 77, 83, 84, 95, 102, 108, 110, 122, 131, 137, 139, 154, 165, 173, 175, 191, 205, 215, 215, 233, 250, 260, 261, 282, 299, 313, 310, 332, 353, 368

Offset: 0

Ilya Gutkovskiy, Oct 25 2022

nonn

Cf. A005117, A087188, A341073, A347586, A347655, A347779, A358023, A358024.

A329069 Expansion of Product_{k>=1} 1 / (1 + mu(k)^2 * x^k).

Original entry on oeis.org

1, -1, 0, -1, 2, -2, 1, -2, 4, -3, 2, -4, 7, -6, 4, -8, 12, -10, 9, -14, 20, -18, 16, -24, 32, -29, 27, -38, 49, -46, 43, -59, 74, -71, 69, -90, 112, -107, 106, -136, 164, -160, 159, -199, 238, -232, 234, -288, 338, -333, 338, -412, 477, -473, 485, -582, 667, -666, 686, -813, 923

Offset: 0

Ilya Gutkovskiy, Nov 04 2019

sign

Convolution inverse of A087188.

Cf. A005117, A008683, A048165, A073576, A087188, A292561.

nmax = 60; CoefficientList[Series[Product[1/(1 + MoebiusMu[k]^2 x^k), {k, 1, nmax}], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[(-1)^(k/d) Boole[SquareFreeQ[d]] d, {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 60}]

G.f.: Product_{k>=1} 1 / (1 + x^A005117(k)).

A331982 Number of compositions (ordered partitions) of n into distinct odd squarefree parts.

Original entry on oeis.org

1, 1, 0, 1, 2, 1, 2, 1, 4, 6, 2, 7, 4, 7, 4, 13, 30, 13, 8, 25, 32, 31, 56, 37, 82, 42, 104, 168, 128, 175, 152, 181, 226, 307, 252, 439, 326, 691, 372, 943, 1190, 1069, 1238, 1435, 2056, 1806, 2102, 2185, 3664, 2550, 4480, 3175, 6090, 3781, 7628, 9691

Offset: 0

Ilya Gutkovskiy, Feb 03 2020

nonn

			a(8) = 4 because we have [7, 1], [5, 3], [3, 5] and [1, 7].

Index entries for sequences related to compositions

Cf. A056911, A087188, A134337, A134345, A280194, A280198, A331846, A331981.

cp's OEIS Frontend

A292561 Expansion of Product_{k>=1} (1 - mu(k)^2*x^k), where mu() is the Moebius function (A008683).

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A280128 Expansion of Product_{k>=2} (1 + mu(k)^2*x^k), where mu(k) is the Moebius function (A008683).

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A300894 L.g.f.: log(Product_{k>=1} (1 + mu(k)^2*x^k)) = Sum_{n>=1} a(n)*x^n/n, where mu() is the Moebius function (A008683).

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A300586 Number of partitions of n into distinct squarefree parts that do not divide n.

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A357525 Expansion of Product_{k>=1} (1 + mu(k)*x^k).

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A358023 Number of partitions of n into at most 2 distinct squarefree parts.

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A358024 Number of partitions of n into at most 3 distinct squarefree parts.

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A358025 Number of partitions of n into at most 4 distinct squarefree parts.

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A329069 Expansion of Product_{k>=1} 1 / (1 + mu(k)^2 * x^k).

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A331982 Number of compositions (ordered partitions) of n into distinct odd squarefree parts.

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A300894 L.g.f.: log(Product_{k>=1} (1 + mu(k)^2x^k)) = Sum_{n>=1} a(n)x^n/n, where mu() is the Moebius function (A008683).