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 A061019 Negate primes in factorization of n. 5

%I

%S 1,-2,-3,4,-5,6,-7,-8,9,10,-11,-12,-13,14,15,16,-17,-18,-19,-20,21,22,

%T -23,24,25,26,-27,-28,-29,-30,-31,-32,33,34,35,36,-37,38,39,40,-41,

%U -42,-43,-44,-45,46,-47,-48,49,-50,51,-52,-53,54,55,56,57,58,-59,60,-61,62,-63,64,65,-66,-67,-68,69,-70

%N Negate primes in factorization of n.

%C Inverse Moebius transform of A001615.

%C a(n) = n*lambda(n), where lambda is Liouville's function: A008836.

%C Totally multiplicative sequence with a(p) = -p for prime p. [_Jaroslav Krizek_, Nov 01 2009]

%F a(n) = (-1)^(number of primes dividing n)*n = n * (-1)^A001222(n) = n*A008836(n).

%F Dirichlet g.f.: zeta(2*s-2)/zeta(s-1). Dirichlet inverse of A055615, all terms turned positive there. - _R. J. Mathar_, Apr 16 2011

%F a(n) = sum_{d|n} lambda(d)*psi(d) = sum_{d|n} A008836(d)* A001615(d) = n/lambda(n). - _Enrique PĂ©rez Herrero_, Sep 18 2012

%e a(6)=(-2)(-3) = +6, while a(8)=(-2)^3 = -8.

%p with(numtheory): P:=proc(n) local p; mul(op(1,p)^op(2,p)*(-1)^op(2,p),p=ifactors(n)[2]); end: seq(P(i),i=1..70); # _Paolo P. Lava_, Mar 22 2018

%t Table[n (-1)^PrimeOmega[n],{n,70}] (* _Harvey P. Dale_, Oct 05 2011 *)

%o a061019 1 = 1

%o a061019 n = product \$ map negate \$ a027746_row n

%o -- _Reinhard Zumkeller_, Feb 08 2012

%o (PARI) a(n) = if( bitand(bigomega(n),1), - n, n ); /* _Joerg Arndt_, Sep 19 2012 */

%Y Cf. A000027, A001222, A061020, A001615.

%Y Cf. A027746.

%Y Cf. A239122 (partial sums).

%K easy,nice,sign,mult

%O 1,2

%A _Marc LeBrun_, Apr 13 2001

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Last modified October 16 18:03 EDT 2019. Contains 328102 sequences. (Running on oeis4.)