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A071068
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Number of ways to write n as a sum of two unordered squarefree numbers.
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23
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0, 1, 1, 2, 1, 2, 2, 3, 2, 2, 2, 4, 3, 3, 3, 5, 4, 4, 3, 6, 4, 5, 4, 7, 5, 5, 5, 7, 5, 5, 5, 8, 6, 7, 6, 11, 7, 7, 7, 11, 8, 8, 9, 13, 10, 8, 8, 13, 10, 8, 7, 14, 10, 10, 7, 13, 10, 11, 9, 15, 11, 11, 11, 15, 11, 11, 11, 18, 12, 13, 11, 21, 13, 14, 13, 20, 14, 13, 14, 20, 16, 13, 13, 22, 15
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OFFSET
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1,4
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COMMENTS
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The natural density of the squarefree numbers is 6/Pi^2, so An < a(n) < Bn for all large enough n with A < 6/Pi^2 - 1/2 and B > 3/Pi^2. The Schnirelmann density of the squarefree numbers is 53/88 > 1/2, and so a(n) > 0 for all n > 1 (in fact, a(n+1) >= 9n/88). It follows from Theoreme 3 bis. in Cohen, Dress, & El Marraki along with finite checking up to 16089908 that 0.10792n < a(n) < 0.303967n for n > 36. (The lower bound holds for n > 1.) - Charles R Greathouse IV, Feb 02 2016
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LINKS
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FORMULA
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a(n) = sum(i=1..floor(n/2), abs(mu(i)*mu(n-i)) ). - Wesley Ivan Hurt, May 20 2013
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EXAMPLE
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12=1+11=2+10=5+7=6+6 hence a(12)=4.
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MATHEMATICA
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Table[Sum[Abs[MoebiusMu[i] MoebiusMu[n - i]], {i, 1, Floor[n/2]}], {n, 1, 85}] (* Indranil Ghosh, Mar 10 2017 *)
Table[Count[IntegerPartitions[n, {2}], _?(AllTrue[#, SquareFreeQ]&)], {n, 90}] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Aug 13 2020 *)
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PROG
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(PARI) list(lim)=my(n=lim\1); concat(0, ceil(Vec((Polrev(vector(n, k, issquarefree(k-1))) + O('x^(n+1)))^2)/2)) \\ Charles R Greathouse IV, May 21 2013
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CROSSREFS
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KEYWORD
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easy,nonn
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AUTHOR
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STATUS
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approved
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