A298473 - OEIS

%I #25 Oct 24 2022 15:12:20

%S 1,-4,-6,8,-10,24,-14,-16,18,40,-22,-48,-26,56,60,32,-34,-72,-38,-80,

%T 84,88,-46,96,50,104,-54,-112,-58,-240,-62,-64,132,136,140,144,-74,

%U 152,156,160,-82,-336,-86,-176,-180,184,-94,-192,98,-200,204,-208,-106,216,220,224,228,232,-118,480

%N a(n) = n * lambda(n) * 2^omega(n).

%C The sequence b(n) = abs(a(n)) = n * 2^omega(n) for n>=1 is multiplicative with b(p^e) = 2*p^e (p prime, e > 0) and is the Dirichlet inverse of a(n). The Dirichlet g.f. of b(n) is: (zeta(s-1))^2/zeta(2*s-2). For omega(n) and lambda(n) see A001221 and A008836, respectively.

%H Robert Israel, <a href="/A298473/b298473.txt">Table of n, a(n) for n = 1..10000</a>

%F Multiplicative with a(p^e) = 2*(-p)^e (p prime, e>0).

%F Dirichlet inverse of abs(a(n)).

%F Dirichlet g.f.: zeta(2*s-2)/(zeta(s-1))^2.

%F Sum_{d|n} A000290(d)*a(n/d) = n*A060648(n).

%F Sum_{d|n} A078439(d)*a(n/d) = A008683(n).

%F O.g.f. for the unsigned sequence: Sum_{n >= 1} |a(n)|*x^n = Sum_{n >= 1} |mu(n)|*n*x^n/(1 - x^n)^2, where mu(n) = A008683(n) is the Möbius function. - _Peter Bala_, Mar 05 2022

%e a(6) = a(2)*a(3) = (-4)*(-6) = 24 = 6*1*2^2;

%e a(8) = a(2^3) = 2*(-2)^3 = -16 = 8*(-1)*2^1.

%p f:= proc(n) local t;

%p mul(2*(-t[1])^t[2],t=ifactors(n)[2])

%p end proc:

%p map(f, [$1..100]); # _Robert Israel_, Mar 06 2022

%t Array[# (-1)^PrimeOmega[#]*2^PrimeNu[#] &, 60] (* _Michael De Vlieger_, Jan 20 2018 *)

%o (PARI) a(n) = n*(-1)^bigomega(n)*2^omega(n); \\ _Michel Marcus_, Jan 20 2018

%Y Cf. A000290, A001221, A001222, A008683, A008836, A060648, A078439.

%K sign,mult

%O 1,2

%A _Werner Schulte_, Jan 19 2018