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A298473
a(n) = n * lambda(n) * 2^omega(n).
5
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
1,2
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.
LINKS
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
EXAMPLE
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.
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 *)
PROG
(PARI) a(n) = n*(-1)^bigomega(n)*2^omega(n); \\ Michel Marcus, Jan 20 2018
KEYWORD
sign,mult
AUTHOR
Werner Schulte, Jan 19 2018
STATUS
approved