A078439 -id:A078439 - cp's OEIS frontend

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

Werner Schulte, Jan 19 2018

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.

			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.

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

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

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

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

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

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

A347104 Dirichlet g.f.: primezeta(s-1) * zeta(s-1) / zeta(s).

Original entry on oeis.org

0, 2, 3, 2, 5, 7, 7, 4, 6, 13, 11, 10, 13, 19, 22, 8, 17, 18, 19, 18, 32, 31, 23, 20, 20, 37, 18, 26, 29, 38, 31, 16, 52, 49, 58, 24, 37, 55, 62, 36, 41, 56, 43, 42, 54, 67, 47, 40, 42, 60, 82, 50, 53, 54, 94, 52, 92, 85, 59, 60, 61, 91, 78, 32, 112, 92, 67, 66, 112, 106, 71, 48, 73, 109, 100

Offset: 1

Ilya Gutkovskiy, Aug 18 2021

nonn

a(n) is the sum of the prime terms in row n of A050873.

Moebius transform of A328260.

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A000010, A001221, A008472, A010051, A018804, A050873, A061397, A078439, A117494, A122411, A255655, A328260, A329354.

Table[DivisorSum[n, MoebiusMu[n/#] # PrimeNu[#] &], {n, 1, 75}]
Table[DivisorSum[n, # EulerPhi[n/#] &, PrimeQ[#] &], {n, 1, 75}]
Table[Sum[Boole[PrimeQ[GCD[n, k]]] GCD[n, k], {k, 1, n}], {n, 1, 75}]

a(n) = sumdiv(n, d, moebius(n/d)*d*omega(d)); \\ Michel Marcus, Aug 18 2021

a(n) = Sum_{d|n} mu(n/d) * d * omega(d).
a(n) = Sum_{p|n, p prime} p * phi(n/p).
a(n) = Sum_{k=1..n} A010051(gcd(n,k)) * gcd(n,k).

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula