1,3

Analog of sigma function A000203[n] with primes negated.

a(n)=sum( d divides n,d*mu(cored(d))) where core(x) is the smallest number such that x*core(x) is a square - Benoit Cloitre (benoit7848c(AT)orange.fr), Apr 07 2002

Unsigned sequence |a(n)| 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 (vladeta(AT)Eunet.yu), Dec 29 2002

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

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 (clary(AT)uakron.edu), Apr 15, 2006

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

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

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

nmax = 72; lambda[k_Integer?Positive] := If[ k > 1, (-1)^Total[ Part[Transpose[FactorInteger[k]], 2] ], 1 ]; Drop[ CoefficientList[ Series[ Sum[ lambda[k] k x^k/(1 - x^k), {k, 1, nmax} ], {x, 0, nmax} ], x ], 1 ] - Stuart Clary (clary(AT)uakron.edu), Apr 15, 2006

(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]) (from R. Stephan)

Cf. A000203, A061019, A076792.

Adjacent sequences: A061017 A061018 A061019 this_sequence A061021 A061022 A061023

Sequence in context: A117744 A091732 A109746 this_sequence A047994 A117009 A103300

easy,sign,mult,mult

Marc LeBrun (mlb(AT)well.com), Apr 13 2001