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A061020 Negate primes in factorizations of divisors of n, then sum. 13
1, -1, -2, 3, -4, 2, -6, -5, 7, 4, -10, -6, -12, 6, 8, 11, -16, -7, -18, -12, 12, 10, -22, 10, 21, 12, -20, -18, -28, -8, -30, -21, 20, 16, 24, 21, -36, 18, 24, 20, -40, -12, -42, -30, -28, 22, -46, -22, 43, -21, 32, -36, -52, 20, 40, 30, 36, 28, -58, 24, -60, 30, -42, 43, 48, -20, -66, -48, 44, -24, -70, -35 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

Analog of sigma function A000203(n) with primes negated.

a(n) = sum( d divides n,d*mu(core(d))) where core(x) = A007913(x) is the smallest number such that x*core(x) is a square. - Benoit Cloitre, Apr 07 2002

Unsigned sequence |a(n)| (A206369) gives the number of numbers 1<=k<=n for which GCD(k,n) is a square. |a(n)| = Sum_{d divides n} d*(-1)^bigomega(n/d). - Vladeta Jovovic, Dec 29 2002

LINKS

T. D. Noe, Table of n, a(n) for n = 1..10000

László Tóth, A survey of the alternating sum-of-divisors function (2011).

FORMULA

Replace each divisor d of n by A061019[d] and sum. Replace p^q with (1-(-p)^(q+1))/(1+p) in prime factorization of n.

Inverse mobius transform of A061019. In other words a(n) = Sum_{d divides n} d*(-1)^bigomega(d), where bigomega(n) = A001222(n).

G.f.: A(x) = sum_{k>=1} lambda(k) k x^k/(1 - x^k) where lambda(k) is the Liouville function, A008836. - Stuart Clary, Apr 15 2006

G.f.: A(x) is x times the logarithmic derivative of A118206(x). - Stuart Clary, Apr 15 2006

Dirichlet g.f.: zeta(s)*zeta(2 s - 2)/zeta(s - 1). - Stuart Clary, Apr 15, 2006

a(n) = Sum(d|n, d*lambda(d))), where lambda(n) is  A008836(n). - Enrique Pérez Herrero, Aug 29 2013

EXAMPLE

a(12) = 1-2-3+4+6-12 = (1-2+4)*(1-3) = -6.

MAPLE

with(numtheory):

A061020 := proc(n) local d; add(d*(-1)^bigomega(d), d=divisors(n)) end:

seq(A061020(n), n=1..72); # Peter Luschny, Aug 29 2013

MATHEMATICA

nmax = 72; Drop[ CoefficientList[ Series[ Sum[ LiouvilleLambda[k] k x^k/(1 - x^k), {k, 1, nmax} ], {x, 0, nmax} ], x ], 1 ] (* Stuart Clary, Apr 15 2006, updated by Jean-François Alcover, Dec 04 2017 *)

PROG

(PARI) for(n=1, 100, print1(sumdiv(n, d, (d)*moebius(core(d))), ", "))

(PARI) a(n)=if(n<1, 0, direuler(p=2, n, 1/(1-X)/(1+p*X))[n]) \\ Ralf Stephan

(PARI) A061020(n) = {my(f=factorint(n)); prod(k=1, #f[, 2], ((-f[k, 1])^(f[k, 2]+1)-1)/(-f[k, 1]-1))} \\ Andrew Lelechenko, Apr 22 2014

(Haskell)

a061020 = sum . map a061019 . a027750_row

-- Reinhard Zumkeller, Feb 08 2012

CROSSREFS

Cf. A000203, A061019, A076792, A206369.

Cf. A027750, A007913.

Sequence in context: A299439 A109746 A286365 * A206369 A152958 A278963

Adjacent sequences:  A061017 A061018 A061019 * A061021 A061022 A061023

KEYWORD

easy,sign,mult

AUTHOR

Marc LeBrun, Apr 13 2001

STATUS

approved

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Last modified August 22 16:37 EDT 2019. Contains 326179 sequences. (Running on oeis4.)