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
Examples
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.
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.
Programs
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Haskell
a192066 = sum . filter odd . a077610_row -- Reinhard Zumkeller, Feb 12 2012
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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:
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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 *)
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PARI
a(n) = sumdiv(n, d, if ((gcd(d, n/d)==1) && (d%2), d)); \\ Michel Marcus, Nov 17 2017
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
Comments