A298473 a(n) = n * lambda(n) * 2^omega(n).

1, -4, -6, 8, -10, 24, -14, -16, 18, 40, -22, -48, -26, 56, 60, 32, -34, -72, -38, -80, 84, 88, -46, 96, 50, 104, -54, -112, -58, -240, -62, -64, 132, 136, 140, 144, -74, 152, 156, 160, -82, -336, -86, -176, -180, 184, -94, -192, 98, -200, 204, -208, -106, 216, 220, 224, 228, 232, -118, 480

Offset: 1

Author

Werner Schulte, Jan 19 2018

Keywords

• sign
• mult

Comments

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.

Examples

			a(6) = a(2)*a(3) = (-4)*(-6) = 24 = 6*1*2^2;
a(8) = a(2^3) = 2*(-2)^3 = -16 = 8*(-1)*2^1.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

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

Programs

Maple

f:= proc(n) local t;
mul(2*(-t[1])^t[2],t=ifactors(n)[2])
end proc:
map(f, [$1..100]); # Robert Israel, Mar 06 2022

Mathematica

Array[# (-1)^PrimeOmega[#]*2^PrimeNu[#] &, 60] (* Michael De Vlieger, Jan 20 2018 *)

PARI

a(n) = n*(-1)^bigomega(n)*2^omega(n); \\ Michel Marcus, Jan 20 2018

Formula

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

Dirichlet inverse of abs(a(n)).

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

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

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

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

Sum_{k=1..n} abs(a(k)) ~ 3*n^2/Pi^2 * (log(n) - 1/2 + 2*gamma - 12*zeta'(2)/Pi^2), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Jul 16 2025