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 A192066 Sum of the odd unitary divisors of n. 2
 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 (list; graph; refs; listen; history; text; internal format)
 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)) ). EXAMPLE 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. 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 *) 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: A192085 A117001 A206787 * A098986 A000593 A115607 Adjacent sequences:  A192063 A192064 A192065 * A192067 A192068 A192069 KEYWORD nonn,mult,easy,changed AUTHOR R. J. Mathar, Jun 22 2011 STATUS approved

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Last modified November 17 14:58 EST 2017. Contains 294834 sequences.