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
Examples
a(12) = 1-2-3+4+6-12 = (1-2+4)*(1-3) = -6.
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.
Programs
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Haskell
a061020 = sum . map a061019 . a027750_row -- Reinhard Zumkeller, Feb 08 2012
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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
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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 *) -
PARI
for(n=1,100,print1(sumdiv(n,d,(d)*moebius(core(d))),","))
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PARI
a(n)=if(n<1,0,direuler(p=2,n,1/(1-X)/(1+p*X))[n]) \\ Ralf Stephan
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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
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
Comments