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 A192066 Sum of the odd unitary divisors of n. 2

%I

%S 1,1,4,1,6,4,8,1,10,6,12,4,14,8,24,1,18,10,20,6,32,12,24,4,26,14,28,8,

%T 30,24,32,1,48,18,48,10,38,20,56,6,42,32,44,12,60,24,48,4,50,26,72,14,

%U 54,28,72,8,80,30,60,24,62,32,80,1,84,48,68,18,96,48,72,10,74,38,104,20,96,56,80,6

%N Sum of the odd unitary divisors of n.

%C The unitary analog of A000593.

%H Reinhard Zumkeller, <a href="/A192066/b192066.txt">Table of n, a(n) for n = 1..10000</a>

%H R. J. Mathar, <a href="http://arxiv.org/abs/1106.4038">Survey of Dirichlet series of multiplicative arithmetic functions</a>, arXiv:1106.4038 [math.NT], 2011-2012, section 4.2.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/UnitaryDivisor.html">Unitary Divisor</a>

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Unitary divisor">Unitary_divisor</a>

%F a(n) = sum_{d|n, d odd, gcd(d,n/d)=1} d.

%F Dirichlet g.f. zeta(s)*zeta(s-1)*(1-2^(1-s))/( zeta(2s-1)*(1-2^(1-2s)) ).

%e n=9 has the divisors 1, 3 and 9, of which 3 is not an unitary divisor because gcd(3,9/3) = gcd(3,3) <>1. This leaves 1 and 9 as unitary divisors which sum to a(9)=1+9=10.

%p unitaryOddSigma := proc(n,k) local a,d ; a := 0 ; for d in numtheory[divisors](n) do if type(d,'odd') then if igcd(d,n/d) = 1 then a := a+d^k ; end if; end if; end do: a ; end proc:

%p A := proc(n) unitaryOddSigma(n,1) ;end proc:

%t a[n_] := DivisorSum[n, Boole[OddQ[#] && GCD[#, n/#] == 1]*#&];

%t Array[a, 80] (* _Jean-François Alcover_, Nov 16 2017 *)

%o a192066 = sum . filter odd . a077610_row

%o -- _Reinhard Zumkeller_, Feb 12 2012

%o (PARI) a(n) = sumdiv(n, d, if ((gcd(d, n/d)==1) && (d%2), d)); \\ _Michel Marcus_, Nov 17 2017

%Y Cf. A068068, A034448.

%Y Cf. A077610, A206787.

%K nonn,mult,easy

%O 1,3

%A _R. J. Mathar_, Jun 22 2011

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