A192066 Sum of the odd unitary divisors of n.

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

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

Cf. A068068, A034448.

Cf. A077610, A206787.

Sequence in context: A117001 A206787 A327419 * A347385 A336113 A098986

Adjacent sequences: A192063 A192064 A192065 * A192067 A192068 A192069

Keyword

nonn,mult,easy

Author

R. J. Mathar, Jun 22 2011

Status

approved