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A076792 Sum_{d divides n} d^2*(-1)^bigomega(d), where bigomega(n) = A001222(n). 2
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

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

LINKS

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

FORMULA

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

EXAMPLE

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

MATHEMATICA

Array[DivisorSum[#, #^2*(-1)^PrimeOmega[#] &] &, 50] (* Michael De Vlieger, Jun 27 2018 *)

PROG

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

(PARI) 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

CROSSREFS

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

Sequence in context: A194427 A185954 A051838 * A146939 A181540 A059028

Adjacent sequences:  A076789 A076790 A076791 * A076793 A076794 A076795

KEYWORD

mult,sign

AUTHOR

Vladeta Jovovic, Nov 16 2002

STATUS

approved

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Last modified March 22 17:25 EDT 2019. Contains 321422 sequences. (Running on oeis4.)