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%I #60 Feb 03 2024 10:15:40
%S 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,
%T 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,
%U 72,8,80,30,60,24,62,32,32,1,84,48,68,18,96,48,72,4
%N Sum of the odd squarefree divisors of n.
%C a(A000079(n)) = 1; a(A057716(n)) > 1; a(A065119(n)) = 4; a(A086761(n)) = 6.
%C 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
%H Reinhard Zumkeller, <a href="/A206787/b206787.txt">Table of n, a(n) for n = 1..10000</a>
%H Jon Maiga, <a href="http://sequencedb.net/s/A206787">Computer-generated formulas for A206787</a>, Sequence Machine.
%F a(n) = Sum_{k = 1..A034444(n)} (A206778(n,k) mod 2) * A206778(n,k).
%F a(n) = Sum_{d|n} d*mu(2*d)^2, where mu is the Möbius function (A008683). - _Ridouane Oudra_, Aug 14 2019
%F Multiplicative with a(2^e) = 1, and a(p^e) = p + 1 for p > 2. - _Amiram Eldar_, Sep 18 2020
%F Sum_{k=1..n} a(k) ~ (1/3) * n^2. - _Amiram Eldar_, Nov 17 2022
%F Dirichlet g.f.: (zeta(s)*zeta(s-1)/zeta(2*s-2))*(2^s/(2^s+2)). - _Amiram Eldar_, Jan 03 2023
%F From _Antti Karttunen_, Nov 22 2023: (Start)
%F a(n) = A000203(A204455(n)) = A000593(A007947(n)) = A048250(n)/A010684(n-1). [From Sequence Machine]
%F a(n) = Sum_{d|n} abs(A349343(d)). [See _R. J. Mathar_'s Jul 12 2012 comment above]
%F (End)
%F a(n) = Sum_{d divides n, d odd} d * mu(d)^2. - _Peter Bala_, Feb 01 2024
%p seq(add(d*mobius(2*d)^2, d in divisors(n)), n=1 .. 80); # _Ridouane Oudra_, Aug 14 2019
%t a[n_] := DivisorSum[n, #*Boole[OddQ[#] && SquareFreeQ[#]]&]; Array[a, 80] (* _Jean-François Alcover_, Dec 05 2015 *)
%t 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 *)
%o (Haskell)
%o a206787 = sum . filter odd . a206778_row
%o (PARI) a(n) = sumdiv(n, d, d*(d % 2)*issquarefree(d)); \\ _Michel Marcus_, Sep 21 2014
%o (Magma) [&+[d:d in Divisors(m)|IsOdd(d) and IsSquarefree(d)]:m in [1..72]]; // _Marius A. Burtea_, Aug 14 2019
%Y Cf. A000203, A000593, A007947, A008683, A056911, A048250, A010684, A192066, A204455.
%Y Cf. A000079, A057716, A065119, A086761.
%Y Inverse Möbius transform of the absolute values of A349343.
%K nonn,mult,easy
%O 1,3
%A _Reinhard Zumkeller_, Feb 12 2012
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