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A206787
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Sum of the odd squarefree divisors of n.
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18
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1, 1, 4, 1, 6, 4, 8, 1, 4, 6, 12, 4, 14, 8, 24, 1, 18, 4, 20, 6, 32, 12, 24, 4, 6, 14, 4, 8, 30, 24, 32, 1, 48, 18, 48, 4, 38, 20, 56, 6, 42, 32, 44, 12, 24, 24, 48, 4, 8, 6, 72, 14, 54, 4, 72, 8, 80, 30, 60, 24, 62, 32, 32, 1, 84, 48, 68, 18, 96, 48, 72, 4
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OFFSET
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1,3
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COMMENTS
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Inverse Mobius transform of 1, 0, 3, 0, 5, 0, 7, 0, 0, 0, 11, 0, 13, 0, 15, 0, 17, 0, 19, 0, 21, 0, 23, 0, 0, 0, 0, 0, 29... - R. J. Mathar, Jul 12 2012
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LINKS
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FORMULA
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Multiplicative with a(2^e) = 1, and a(p^e) = p + 1 for p > 2. - Amiram Eldar, Sep 18 2020
Sum_{k=1..n} a(k) ~ (1/3) * n^2. - Amiram Eldar, Nov 17 2022
Dirichlet g.f.: (zeta(s)*zeta(s-1)/zeta(2*s-2))*(2^s/(2^s+2)). - Amiram Eldar, Jan 03 2023
(End)
a(n) = Sum_{d divides n, d odd} d * mu(d)^2. - Peter Bala, Feb 01 2024
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MAPLE
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seq(add(d*mobius(2*d)^2, d in divisors(n)), n=1 .. 80); # Ridouane Oudra, Aug 14 2019
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MATHEMATICA
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a[n_] := DivisorSum[n, #*Boole[OddQ[#] && SquareFreeQ[#]]&]; Array[a, 80] (* Jean-François Alcover, Dec 05 2015 *)
f[2, e_] := 1; f[p_, e_] := p + 1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 18 2020 *)
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PROG
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(Haskell)
a206787 = sum . filter odd . a206778_row
(PARI) a(n) = sumdiv(n, d, d*(d % 2)*issquarefree(d)); \\ Michel Marcus, Sep 21 2014
(Magma) [&+[d:d in Divisors(m)|IsOdd(d) and IsSquarefree(d)]:m in [1..72]]; // Marius A. Burtea, Aug 14 2019
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CROSSREFS
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Inverse Möbius transform of the absolute values of A349343.
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KEYWORD
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nonn,mult,easy
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AUTHOR
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STATUS
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approved
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