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A061019 Negate primes in factorization of n. 5
1, -2, -3, 4, -5, 6, -7, -8, 9, 10, -11, -12, -13, 14, 15, 16, -17, -18, -19, -20, 21, 22, -23, 24, 25, 26, -27, -28, -29, -30, -31, -32, 33, 34, 35, 36, -37, 38, 39, 40, -41, -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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Inverse Moebius transform of A001615.

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

Totally multiplicative sequence with a(p) = -p for prime p. [Jaroslav Krizek, Nov 01 2009]

LINKS

Table of n, a(n) for n=1..70.

FORMULA

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

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

a(n) = sum_{d|n} lambda(d)*psi(d) = sum_{d|n} A008836(d)* A001615(d) = n/lambda(n). -  Enrique Pérez Herrero, Sep 18 2012

EXAMPLE

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

MAPLE

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

MATHEMATICA

Table[n (-1)^PrimeOmega[n], {n, 70}] (* Harvey P. Dale, Oct 05 2011 *)

PROG

(Haskell)

a061019 1 = 1

a061019 n = product $ map negate $ a027746_row n

-- Reinhard Zumkeller, Feb 08 2012

(PARI) a(n) = if( bitand(bigomega(n), 1), - n, n ); /* Joerg Arndt, Sep 19 2012 */

CROSSREFS

Cf. A000027, A001222, A061020, A001615.

Cf. A027746.

Cf. A239122 (partial sums).

Sequence in context: A069782 A088480 A289207 * A028310 A097045 A118760

Adjacent sequences:  A061016 A061017 A061018 * A061020 A061021 A061022

KEYWORD

easy,nice,sign,mult

AUTHOR

Marc LeBrun, Apr 13 2001

STATUS

approved

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Last modified July 22 10:41 EDT 2019. Contains 325219 sequences. (Running on oeis4.)