OFFSET
1,3
COMMENTS
The unitary analog of A000593.
LINKS
Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
R. J. Mathar, Survey of Dirichlet series of multiplicative arithmetic functions, arXiv:1106.4038 [math.NT], 2011-2012, section 4.2.
Eric Weisstein's World of Mathematics, Unitary Divisor.
Wikipedia, Unitary divisor.
FORMULA
a(n) = Sum_{d|n, d odd, gcd(d,n/d)=1} d.
Dirichlet g.f.: zeta(s)*zeta(s-1)*(1-2^(1-s))/( zeta(2s-1)*(1-2^(1-2s)) ).
Sum_{k=1..n} a(k) ~ Pi^2 * n^2 / (21*zeta(3)). - Vaclav Kotesovec, Feb 02 2019
Multiplicative with a(2^e) = 1, and a(p^e) = p^e + 1 for p > 2. - Amiram Eldar, Sep 18 2020
EXAMPLE
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.
MAPLE
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:
A := proc(n) unitaryOddSigma(n, 1) ; end proc:
MATHEMATICA
a[n_] := DivisorSum[n, Boole[OddQ[#] && GCD[#, n/#] == 1]*#&];
Array[a, 80] (* Jean-François Alcover, Nov 16 2017 *)
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 *)
PROG
(Haskell)
a192066 = sum . filter odd . a077610_row
-- Reinhard Zumkeller, Feb 12 2012
(PARI) a(n) = sumdiv(n, d, if ((gcd(d, n/d)==1) && (d%2), d)); \\ Michel Marcus, Nov 17 2017
CROSSREFS
KEYWORD
nonn,mult,easy
AUTHOR
R. J. Mathar, Jun 22 2011
STATUS
approved