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A073576
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Number of partitions of n into squarefree parts.
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45
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1, 1, 2, 3, 4, 6, 9, 12, 16, 21, 28, 36, 47, 60, 76, 96, 120, 150, 185, 228, 280, 342, 416, 504, 608, 731, 877, 1048, 1249, 1484, 1759, 2079, 2452, 2885, 3387, 3968, 4640, 5413, 6304, 7328, 8504, 9852, 11395, 13159, 15172, 17468, 20082, 23056, 26434, 30267
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OFFSET
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0,3
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COMMENTS
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Euler transform of the absolute values of A008683. - Tilman Neumann, Dec 13 2008
Euler transform of A008966. - Vaclav Kotesovec, Mar 31 2018
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LINKS
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Alois P. Heinz, Table of n, a(n) for n = 0..10000
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FORMULA
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G.f.: 1/Product_{k>0} (1-x^A005117(k)).
a(n) = 1/n*Sum_{k=1..n} A048250(k)*a(n-k).
a(n) = A000041(n) - A114374(n) - A117395(n), n>0. - Reinhard Zumkeller, Mar 11 2006
G.f.: 1 + Sum_{i>=1} mu(i)^2*x^i / Product_{j=1..i} (1 - mu(j)^2*x^j). - Ilya Gutkovskiy, Jun 05 2017
a(n) ~ exp(2*sqrt(n)) / (4*Pi^(3/2)*n^(1/4)). - Vaclav Kotesovec, Mar 24 2018
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MAPLE
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with(numtheory):
a:= proc(n) option remember; `if`(n=0, 1, add(add(d*
abs(mobius(d)), d=divisors(j)) *a(n-j), j=1..n)/n)
end:
seq(a(n), n=0..60); # Alois P. Heinz, Mar 05 2015
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MATHEMATICA
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Table[Length[Select[Boole /@ Thread /@ SquareFreeQ /@ IntegerPartitions[n], FreeQ[#, 0] &]], {n, 48}] (* Jayanta Basu, Jul 02 2013 *)
a[n_] := a[n] = If[n==0, 1, Sum[Sum[d*Abs[MoebiusMu[d]], {d, Divisors[j]}] * a[n-j], {j, 1, n}]/n]; Table[a[n], {n, 0, 60}] (* Jean-François Alcover, Oct 10 2015, after Alois P. Heinz *)
nmax = 60; CoefficientList[Series[Exp[Sum[Sum[Abs[MoebiusMu[k]] * x^(j*k) / j, {k, 1, Floor[nmax/j] + 1}], {j, 1, nmax}]], {x, 0, nmax}], x] (* Vaclav Kotesovec, Mar 31 2018 *)
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PROG
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(Haskell)
a073576 = p a005117_list where
p _ 0 = 1
p ks'@(k:ks) m = if m < k then 0 else p ks' (m - k) + p ks m
-- Reinhard Zumkeller, Jun 01 2015
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CROSSREFS
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Cf. A058647.
Cf. A087188.
Cf. A225244.
Cf. A114374.
Sequence in context: A060729 A229169 A155510 * A186115 A187020 A058647
Adjacent sequences: A073573 A073574 A073575 * A073577 A073578 A073579
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KEYWORD
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nonn
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AUTHOR
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Vladeta Jovovic, Aug 27 2002
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STATUS
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approved
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