|
%I #36 Aug 27 2023 04:22:45
%S 1,-3,-8,13,-24,24,-48,-51,73,72,-120,-104,-168,144,192,205,-288,-219,
%T -360,-312,384,360,-528,408,601,504,-656,-624,-840,-576,-960,-819,960,
%U 864,1152,949,-1368,1080,1344,1224,-1680,-1152,-1848,-1560,-1752,1584,-2208,-1640,2353,-1803
%N Sum_{d divides n} d^2*(-1)^bigomega(d), where bigomega(n) = A001222(n).
%C The sign of a(n) is (-1)^(bigomega(n)) = (-1)^(A001222(n)). - _David A. Corneth_, Jun 27 2018
%H Daniel Suteu, <a href="/A076792/b076792.txt">Table of n, a(n) for n = 1..10000</a>
%F Multiplicative with a(p^e) = (1+(-1)^e*p^(2*e+2))/(1+p^2).
%F Dirichlet g.f.: zeta(s)*zeta(2*s-4)/zeta(s-2).
%F 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).
%F Dirichlet g.f. for b(n, k): zeta(s)*zeta(2*s-2*k)/zeta(s-k).
%F b(n, 0) = A010052(n), b(n, 1) = A061020(n).
%F a(n) = A008836(n)*n^2* Sum(d|n, A008836(d)/d^2). - _Enrique Pérez Herrero_, Jul 10 2012
%F a(n) = (-1)^bigomega(n) * Sum_{d|n, d is a perfect square} A007434(n/d). - _Daniel Suteu_, Jun 27 2018
%F Sum_{k=1..n} |a(k)| ~ n^3 * zeta(6)/(3*zeta(3)). - _Daniel Suteu_, Apr 06 2019
%F Dirichlet g.f. for |a(n)|: zeta(s-2)*zeta(2*s)/zeta(s). - _Vaclav Kotesovec_, Apr 06 2019
%e 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
%t Array[DivisorSum[#, #^2*(-1)^PrimeOmega[#] &] &, 50] (* _Michael De Vlieger_, Jun 27 2018 *)
%t 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 *)
%o (PARI) a(n) = sumdiv(n, d, d^2 * (-1)^bigomega(d)); \\ _Daniel Suteu_, Jun 27 2018
%o (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
%Y Cf. A001222, A007434, A008836, A010052, A061020.
%K mult,sign,easy
%O 1,2
%A _Vladeta Jovovic_, Nov 16 2002
|
