A192066 - OEIS

%I #34 Jul 18 2022 22:48:27

%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)) ).

%F Sum_{k=1..n} a(k) ~ Pi^2 * n^2 / (21*zeta(3)). - _Vaclav Kotesovec_, Feb 02 2019

%F Multiplicative with a(2^e) = 1, and a(p^e) = p^e + 1 for p > 2. - _Amiram Eldar_, Sep 18 2020

%e n=9 has the divisors 1, 3 and 9, of which 3 is not a 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 *)

%t f[2, p_] := 1; f[p_, e_] := p^e + 1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* _Amiram Eldar_, Sep 18 2020 *)

%o (Haskell)

%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