A061020 Negate primes in factorizations of divisors of n, then sum.

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

Offset

1,3

Comments

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

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|n} d*(-1)^bigomega(n/d). - Vladeta Jovovic, Dec 29 2002

Links

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

Dana G. Korssjoen, Biyao Li, Stefan Steinerberger, Raghavendra Tripathi, and Ruimin Zhang, Finding structure in sequences of real numbers via graph theory: a problem list, Involve, a Journal of Mathematics, Vol. 15, No. 2 (2022), pp. 251-270; arXiv preprint, arXiv:2012.04625 [math.CO], 2020-2021.

László Tóth, A survey of the alternating sum-of-divisors function, arXiv:1111.4842 [math.NT], 2011-2014.

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|n} d*(-1)^bigomega(d), where bigomega(n) = A001222(n).

a(n) = Sum_{d|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

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 *)
f[p_, e_] := ((-p)^(e+1)-1)/(-p-1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 24 2023 *)

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: A109746 A286365 A345061 * A206369 A152958 A278963

Adjacent sequences: A061017 A061018 A061019 * A061021 A061022 A061023

Keyword

easy,sign,mult

Author

Marc LeBrun, Apr 13 2001

Status

approved