A061020 -id:A061020 - cp's OEIS frontend

A076792 Sum_{d divides n} d^2*(-1)^bigomega(d), where bigomega(n) = A001222(n).

1, -3, -8, 13, -24, 24, -48, -51, 73, 72, -120, -104, -168, 144, 192, 205, -288, -219, -360, -312, 384, 360, -528, 408, 601, 504, -656, -624, -840, -576, -960, -819, 960, 864, 1152, 949, -1368, 1080, 1344, 1224, -1680, -1152, -1848, -1560, -1752, 1584, -2208, -1640, 2353, -1803

Offset: 1

Vladeta Jovovic, Nov 16 2002

The sign of a(n) is (-1)^(bigomega(n)) = (-1)^(A001222(n)). - David A. Corneth, Jun 27 2018

			As 12 = 2^2 * 3, a(12) = a(2^2) * a(3) = (1+(-1)^2*2^(2*2+2))/(1+2^2) * (1+(-1)^1*3^(2*1+2))/(1+3^2) = 13 * -8 = -104. - _David A. Corneth_, Jun 27 2018

Daniel Suteu, Table of n, a(n) for n = 1..10000

Cf. A001222, A007434, A008836, A010052, A061020.

Array[DivisorSum[#, #^2*(-1)^PrimeOmega[#] &] &, 50] (* Michael De Vlieger, Jun 27 2018 *)
f[p_, e_] := (1 + (-1)^e*p^(2*e+2))/(1 + p^2); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Aug 27 2023 *)

a(n) = sumdiv(n, d, d^2 * (-1)^bigomega(d)); \\ Daniel Suteu, Jun 27 2018

a(n) = my(f=factor(n)); prod(k=1, #f~, ((-1)^f[k,2] * f[k,1]^(2 * f[k,2] + 2) + 1) / (1 + f[k,1]^2)); \\ Daniel Suteu, Jun 27 2018

Multiplicative with a(p^e) = (1+(-1)^e*p^(2*e+2))/(1+p^2).
Dirichlet g.f.: zeta(s)*zeta(2*s-4)/zeta(s-2).
More generally, if b(n, k) = Sum_{d divides n} d^k*(-1)^bigomega(d) then b(n, k) is multiplicative and b(p^e, k) = (1+(-1)^e*p^(k*(e+1)))/(1+p^k).
Dirichlet g.f. for b(n, k): zeta(s)*zeta(2*s-2*k)/zeta(s-k).
b(n, 0) = A010052(n), b(n, 1) = A061020(n).
a(n) = A008836(n)*n^2* Sum(d|n, A008836(d)/d^2). - Enrique Pérez Herrero, Jul 10 2012
a(n) = (-1)^bigomega(n) * Sum_{d|n, d is a perfect square} A007434(n/d). - Daniel Suteu, Jun 27 2018
Sum_{k=1..n} |a(k)| ~ n^3 * zeta(6)/(3*zeta(3)). - Daniel Suteu, Apr 06 2019
Dirichlet g.f. for |a(n)|: zeta(s-2)*zeta(2*s)/zeta(s). - Vaclav Kotesovec, Apr 06 2019

A118209 Expansion of Sum_{k>=1} lambda(k) * k * x^k/(1 + x^k) where lambda(k) is the Liouville function, A008836.

Original entry on oeis.org

1, -3, -2, 5, -4, 6, -6, -11, 7, 12, -10, -10, -12, 18, 8, 21, -16, -21, -18, -20, 12, 30, -22, 22, 21, 36, -20, -30, -28, -24, -30, -43, 20, 48, 24, 35, -36, 54, 24, 44, -40, -36, -42, -50, -28, 66, -46, -42, 43, -63, 32, -60, -52, 60, 40, 66, 36, 84, -58, 40, -60, 90, -42, 85, 48, -60, -66, -80, 44, -72, -70, -77, -72, 108, -42

Offset: 1

Stuart Clary, Apr 15 2006

Related to the logarithmic derivative of A118207(x) and A118208(x).

Related to a signed variant of A022998 via Mobius inversion. - R. J. Mathar, Jul 03 2011

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A008836, A022998, A028260, A061019, A061020, A062157, A117212, A118207, A118208.

nmax = 80; lambda[k_Integer?Positive] := If[ k > 1, (-1)^Total[ Part[Transpose[FactorInteger[k]], 2] ], 1 ]; Drop[ CoefficientList[ Series[ Sum[ lambda[k] k x^k/(1 + x^k), {k, 1, nmax} ], {x, 0, nmax} ], x ], 1 ]
f[p_, e_] := (p*(-p)^e+1)/(p+1); f[2, e_] := ((-1)^e*2^(e+2) - 1)/3; a[n_] := Times @@ f @@@ FactorInteger[n]; a[1] = 1; Array[a, 100] (* Amiram Eldar, Aug 12 2023 *)

a(n) = sumdiv(n, d, (-1)^(n/d - 1)*(-1)^vecsum(factor(d)[,2])*d) \\ Michel Marcus, Dec 10 2016

a(n) = Sum_{d|n} (-1)^(n/d - 1)*lambda(d)*d, Dirichlet convolution of A061019 and A062157.
G.f.: A(x) is x times the logarithmic derivative of A118207(x).
G.f.: A(x) = A061020(x) - 2 A061020(x^2).
Dirichlet g.f.: zeta(s)*zeta(2s-2)*(1-2^(1-s))/zeta(s-1). - R. J. Mathar, Jul 03 2011
a(n) > 0 for n in A028260. - Michel Marcus, Dec 10 2016
Multiplicative with a(2^e) = ((-1)^e*2^(e+2) - 1)/3, and a(p^e) = (p*(-p)^e+1)/(p+1) for an odd prime p. - Amiram Eldar, Aug 12 2023

A328181 a(n) = (-1)^(bigomega(n) - omega(n)) * Sum_{d|n} (-1)^(bigomega(d) - omega(d)) * d.

Original entry on oeis.org

1, 3, 4, 1, 6, 12, 8, 7, 5, 18, 12, 4, 14, 24, 24, 9, 18, 15, 20, 6, 32, 36, 24, 28, 19, 42, 22, 8, 30, 72, 32, 23, 48, 54, 48, 5, 38, 60, 56, 42, 42, 96, 44, 12, 30, 72, 48, 36, 41, 57, 72, 14, 54, 66, 72, 56, 80, 90, 60, 24, 62, 96, 40, 41, 84, 144, 68, 18, 96, 144

Offset: 1

Ilya Gutkovskiy, Oct 06 2019

Amiram Eldar, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Distinct Prime Factors.
Eric Weisstein's World of Mathematics, Prime Factor.

Cf. A001221, A001222, A008836, A046660, A049060, A055076, A061020, A076479, A162511.

a[n_] := (-1)^(PrimeOmega[n] - PrimeNu[n]) Sum[(-1)^(PrimeOmega[d] - PrimeNu[d]) d, {d, Divisors[n]}]; Table[a[n], {n, 1, 70}]
f[p_, e_] := (p^(e+1) - (-1)^e *(2*p+1))/(p+1); a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Dec 02 2020 *)

a(n) = (-1)^(bigomega(n)-omega(n))*sumdiv(n, d, (-1)^(bigomega(d)-omega(d))*d); \\ Michel Marcus, Oct 06 2019

a(p) = p + 1, where p is prime.
Multiplicative with a(p^e) = (p^(e+1) - (-1)^e*(2*p+1))/(p+1). - Amiram Eldar, Dec 02 2020
Sum_{k=1..n} a(k) ~ c * n^2, where c = (Pi^2/30) * Product_{p prime} (1 + 2/p^2 - 2/p^3) = 0.5507877576... . - Amiram Eldar, Nov 06 2022

A200758 Superimperfect numbers.

Original entry on oeis.org

2, 4, 8, 128, 32768, 2147483648

Offset: 1

Laszlo Toth, Nov 22 2011

A number n is said to be superimperfect if 2*beta(beta(n)) = n, where beta is the multiplicative function defined by beta(p^e) = p^e - p^(e-1) + p^(e-2) - ... + (-1)^e for every prime power p^e. The function beta is called the alternating sum-of-divisors function. Here beta(n) is the absolute value of A061020(n). There are no other superimperfect numbers up to 10^7. The number 2^(2^k-1) is superimperfect if and only if k=1,2,3,4,5.

Laszlo Toth, A survey of the alternating sum-of-divisors function, arXiv:1111.4842 [math.NT], 2011-2014.
Laszlo Toth, The alternating sum-of-divisors function, 9th Joint Conf. on Math. and Comp. Sci., February 9-12, 2012, Siofok, Hungary.

Cf. A061020, A127724, A127725, A019279, A058891, A206369.

beta(n)=sumdiv(n,d,(-1)^bigomega(n/d)*d)
for(n=1,1e8,if(2*beta(beta(n))==n,print1(n", "))) \\ Charles R Greathouse IV, Nov 22 2011

ak(p,e)=my(s=1); for(i=1,e, s=s*p + (-1)^i); s
beta(n)=my(f=factor(n)); prod(i=1,#f~, ak(f[i,1],f[i,2]))
is(n)=my(b=beta(n)); 2*b-2 >= n && 2*beta(b)==n \\ Charles R Greathouse IV, Dec 27 2016

A307837 a(1) = 1; a(n+1) = Sum_{d|n} lambda(d)*a(d), where lambda = Liouville function (A008836).

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 1, 0, 0, 1, 0, 1, 1, 0, -1, -1, -1, 2, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 1, 2, 1, 0, 1, 0, 2, -1, -2, 3, -3, -2, 1, 1, -1, 2, 3, 3, 2, 3, 3, -2, -3, 4, 4, -3, -3, -3, 4, -3, 4, 4, -3, 4, -5, 6, 6, -6, 8, 9, -9, 10, -8, -6, -7, 8, 7, 6, 5, 6, 7, -6, -8, -7, 6, 7, 9, 9, 5, -4, 2, -1

Offset: 1

Ilya Gutkovskiy, May 01 2019

sign

Cf. A001222, A008836, A061020, A070965, A206369.

a[n_] := a[n] = Sum[LiouvilleLambda[d] a[d], {d, Divisors[n - 1]}]; a[1] = 1; Table[a[n], {n, 1, 100}]
a[n_] := a[n] = SeriesCoefficient[x (1 + Sum[LiouvilleLambda[k] a[k] x^k/(1 - x^k), {k, 1, n - 1}]), {x, 0, n}]; Table[a[n], {n, 1, 100}]

G.f.: x * (1 + Sum_{n>=1} lambda(n)*a(n)*x^n/(1 - x^n)).

cp's OEIS Frontend

A076792 Sum_{d divides n} d^2*(-1)^bigomega(d), where bigomega(n) = A001222(n).

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A118209 Expansion of Sum_{k>=1} lambda(k) * k * x^k/(1 + x^k) where lambda(k) is the Liouville function, A008836.

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A328181 a(n) = (-1)^(bigomega(n) - omega(n)) * Sum_{d|n} (-1)^(bigomega(d) - omega(d)) * d.

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A200758 Superimperfect numbers.

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A307837 a(1) = 1; a(n+1) = Sum_{d|n} lambda(d)*a(d), where lambda = Liouville function (A008836).

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