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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], 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:

PROG

(Haskell)

a192066 = sum . filter odd . a077610_row

-- Reinhard Zumkeller, Feb 12 2012

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

AUTHOR

R. J. Mathar, Jun 22 2011

STATUS

approved

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Last modified August 28 20:16 EDT 2015. Contains 261164 sequences.