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A369188
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Number of squarefree triangular divisors of n.
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2
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1, 1, 2, 1, 1, 3, 1, 1, 2, 2, 1, 3, 1, 1, 3, 1, 1, 3, 1, 2, 3, 1, 1, 3, 1, 1, 2, 1, 1, 5, 1, 1, 2, 1, 1, 3, 1, 1, 2, 2, 1, 4, 1, 1, 3, 1, 1, 3, 1, 2, 2, 1, 1, 3, 2, 1, 2, 1, 1, 5, 1, 1, 3, 1, 1, 4, 1, 1, 2, 2, 1, 3, 1, 1, 3, 1, 1, 4, 1, 2, 2, 1, 1, 4, 1, 1, 2, 1, 1, 5
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OFFSET
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1,3
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COMMENTS
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Inverse Möbius transform of mu(n)^2 * c(n), where c(n) is the characteristic function of triangular numbers (A010054). - Wesley Ivan Hurt, Jun 21 2024
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LINKS
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FORMULA
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a(n) = Sum_{d|n} mu(d)^2 * c(d), where c = A010054.
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Sum_{k>=1} 1/A061304(k) = 1.83695021... . - Amiram Eldar, Jan 20 2024
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MATHEMATICA
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Table[Sum[MoebiusMu[d]^2 (Floor[Sqrt[2 d + 1] + 1/2] - Floor[Sqrt[2 d] + 1/2]), {d, Divisors[n]}], {n, 100}]
a[n_] := DivisorSum[n, 1 &, IntegerQ@ Sqrt[8*# + 1] && SquareFreeQ[#] &]; Array[a, 100] (* Amiram Eldar, Jan 20 2024 *)
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PROG
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(PARI) a(n) = sumdiv(n, d, issquare(8*d+1) && issquarefree(d)); \\ Amiram Eldar, Jan 20 2024
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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