A118206 -id:A118206 - cp's OEIS frontend

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

Marc LeBrun, Apr 13 2001

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

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

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.

Cf. A000203, A061019, A076792, A206369.

Cf. A027750, A007913.

a061020 = sum . map a061019 . a027750_row
-- Reinhard Zumkeller, Feb 08 2012

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

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 *)

for(n=1,100,print1(sumdiv(n,d,(d)*moebius(core(d))),","))

a(n)=if(n<1,0,direuler(p=2,n,1/(1-X)/(1+p*X))[n]) \\ Ralf Stephan

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

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

A118207 Expansion of Product_{k>=1} (1 + x^k)^lambda(k) where lambda(k) is the Liouville function, A008836.

Original entry on oeis.org

1, 1, -1, -2, 1, 2, 0, -2, -2, 0, 5, 2, -7, -6, 7, 9, 0, -10, -9, 4, 17, 2, -18, -12, 14, 21, 5, -26, -25, 14, 41, 4, -38, -35, 18, 53, 23, -56, -54, 31, 86, 15, -78, -85, 34, 112, 41, -110, -102, 49, 158, 40, -138, -150, 68, 195, 68, -191, -190, 69, 279, 89, -217, -253, 102, 327, 122, -336, -335, 118, 462, 142, -361, -430, 170

Offset: 0

Stuart Clary, Apr 15 2006

Wikipedia, Cyclotomic polynomial
Wikipedia, Liouville function

Cf. A117210, A118205, A118206, A118208, A118209.

nmax = 80; lambda[k_Integer?Positive] := If[ k > 1, (-1)^Total[ Part[Transpose[FactorInteger[k]], 2] ], 1 ]; CoefficientList[ Series[ Product[ (1 + x^k)^lambda[k], {k, 1, nmax} ], {x, 0, nmax} ], x ]
(* From version 7 on *) nmax = 80; CoefficientList[ Series[ Product[ (1 + x^k)^LiouvilleLambda[k], {k, 1, nmax}], {x, 0, nmax}], x] (* Jean-François Alcover, Jul 30 2013 *)

From Peter Bala, Apr 05 2023: (Start)
G.f.: A(x) = Product_{k >= 1} C(k,x^(2*k)) / C(k,x^k) = Product_{k >= 1} C(2*k,x^k) / C(4*k,x^k) = -Product_{k >= 1} C(k,x^(2*k)) * C(2*k,x^k), where C(k,x) denotes the k-th cyclotomic polynomial.
Conjecture: A(x^2) = Product_{k >= 1} C(k,x^k) * C(k,(-x)^k). (End)

A118205 Euler transform of the negative of the Liouville function.

Original entry on oeis.org

1, -1, 1, 0, -1, 2, -2, 2, 0, -2, 3, -2, 1, 2, -3, 3, -2, 0, 3, -2, 3, -2, 0, 2, -2, 3, -1, 0, 1, -2, 5, 0, 0, 1, -2, 1, 1, 2, 0, 1, -2, 1, 4, -1, 4, -2, -3, 6, -2, 5, 6, -8, 6, -4, 2, 9, -8, 7, -4, -1, 11, -1, 5, 1, -8, 5, 2, 4, 7, -8, 4, 2, 1, 14, -2, 0, -1, -6, 19, 2, 5, 6, -15, 12, 1, 3, 18, -17, 1, 9, 0, 29, -4, -3, 4, -13, 14, 17, 2, 0, -4

Offset: 0

Stuart Clary, Apr 15 2006

Peter Bala, Cyclotomic polynomials and the generating functions of A118205 and A118206
Wikipedia, Cyclotomic polynomial
Wikipedia, Liouville function

Cf. A061020, A117208, A118206, A118207, A118208.

nmax = 100; lambda[k_Integer?Positive] := If[ k > 1, (-1)^Total[ Part[Transpose[FactorInteger[k]], 2] ], 1 ]; CoefficientList[ Series[ Product[ (1 - x^k)^lambda[k], {k, 1, nmax} ], {x, 0, nmax} ], x ]
(* Second program (needs Mma >= 7.0): *)
nmax = 100;
Product[(1 - x^n)^LiouvilleLambda[n], {n, 1, nmax}] + O[x]^(nmax+1) // CoefficientList[#, x]& (* Jean-François Alcover, Jan 08 2020 *)

G.f.: A(x) = Product_{k>=1} (1 - x^k)^lambda(k) where lambda(k) is the Liouville function, A008836.

G.f.: A(x) = - Product_{k >= 1} C(k,x^k), where C(k,x) denotes the k-th cyclotomic polynomial. - Peter Bala, Mar 31 2023

A118208 G.f.: A(x) = Product_{k>=1} (1 + x^k)^(-lambda(k)) where lambda(k) is the Liouville function, A008836.

Original entry on oeis.org

1, -1, 2, -1, 0, 2, -4, 5, -3, 0, 4, -6, 6, -2, -3, 8, -10, 6, 0, -6, 14, -13, 9, 0, -12, 17, -18, 11, 3, -18, 28, -22, 14, 7, -25, 30, -31, 11, 12, -23, 34, -28, 9, 12, -30, 35, -31, 10, 11, -30, 56, -35, 26, -4, -41, 51, -65, 48, -8, -28, 65, -74, 70, -9, -49, 71, -112, 69, -4, -48, 135, -129, 82, -21, -83, 155, -176, 99, 0

Offset: 0

Stuart Clary, Apr 15 2006

Wikipedia, Cyclotomic polynomial
Wikipedia, Liouville function

Cf. A118205, A118206, A118207, A118209, A117211.

nmax = 80; lambda[k_Integer?Positive] := If[ k > 1, (-1)^Total[ Part[Transpose[FactorInteger[k]], 2] ], 1 ]; CoefficientList[ Series[ Product[ (1 + x^k)^(-lambda[k]), {k, 1, nmax} ], {x, 0, nmax} ], x ]

G.f.: A(x) = Product_{k >= 1} C(k,x^k)*C(2*k,x^(2*k)), where C(k,x) denotes the k-th cyclotomic polynomial. - Peter Bala, Mar 31 2023

A307076 Expansion of 1/(1 - Sum_{k>=1} lambda(k)*x^k), where lambda() is the Liouville function (A008836).

Original entry on oeis.org

1, 1, 0, -2, -2, 0, 4, 4, -2, -10, -6, 10, 22, 4, -34, -46, 16, 102, 86, -100, -272, -126, 370, 650, 60, -1138, -1384, 526, 3142, 2532, -2936, -7952, -3440, 10802, 18426, 596, -33344, -38418, 18716, 91934, 68400, -93402, -230962, -86236, 330144, 528880, -17298, -996040

Offset: 0

Ilya Gutkovskiy, Mar 22 2019

sign

Invert transform of A008836.

Cf. A008836, A118206, A118207, A300663.

nmax = 47; CoefficientList[Series[1/(1 - Sum[LiouvilleLambda[k] x^k, {k, 1, nmax}]), {x, 0, nmax}], x]
a[0] = 1; a[n_] := a[n] = Sum[LiouvilleLambda[k] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 47}]

a(0) = 1; a(n) = Sum_{k=1..n} A008836(k)*a(n-k).

A356907 Expansion of 1 / (1 + Sum_{k>=1} lambda(k)*x^k), where lambda() is the Liouville function (A008836).

Original entry on oeis.org

1, -1, 2, -2, 2, 0, -4, 12, -22, 34, -42, 38, -6, -68, 202, -394, 616, -782, 730, -204, -1104, 3486, -6994, 11142, -14452, 14026, -5296, -17558, 60042, -123860, 201128, -266384, 268176, -124034, -273626, 1030396, -2188864, 3624290, -4898740, 5101306, -2744408

Offset: 0

Ilya Gutkovskiy, Oct 02 2022

sign

Cf. A008836, A118205, A118206, A118207, A118208, A185694, A307076, A307240.

nmax = 40; CoefficientList[Series[1/(1 + Sum[LiouvilleLambda[k] x^k, {k, 1, nmax}]), {x, 0, nmax}], x]
a[0] = 1; a[n_] := a[n] = -Sum[LiouvilleLambda[k] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 40}]

a(0) = 1; a(n) = -Sum_{k=1..n} A008836(k) * a(n-k).

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

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A118207 Expansion of Product_{k>=1} (1 + x^k)^lambda(k) where lambda(k) is the Liouville function, A008836.

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A118205 Euler transform of the negative of the Liouville function.

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A118208 G.f.: A(x) = Product_{k>=1} (1 + x^k)^(-lambda(k)) where lambda(k) is the Liouville function, A008836.

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A307076 Expansion of 1/(1 - Sum_{k>=1} lambda(k)*x^k), where lambda() is the Liouville function (A008836).

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A356907 Expansion of 1 / (1 + Sum_{k>=1} lambda(k)*x^k), where lambda() is the Liouville function (A008836).

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