A338164 -id:A338164 - cp's OEIS frontend

A360428 Inverse Mobius transformation of A338164.

1, 7, 17, 40, 49, 119, 97, 208, 225, 343, 241, 680, 337, 679, 833, 1024, 577, 1575, 721, 1960, 1649, 1687, 1057, 3536, 1825, 2359, 2673, 3880, 1681, 5831, 1921, 4864, 4097, 4039, 4753, 9000, 2737, 5047, 5729, 10192, 3361, 11543, 3697, 9640, 11025, 7399, 4417, 17408, 7105, 12775

Offset: 1

R. J. Mathar, Feb 07 2023

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000005, A001620, A007434, A008683, A018804, A069097, A338164.

A360428 := proc(n)
    add(numtheory[mobius](n/d)*numtheory[tau](d)*d^2,d=numtheory[divisors](n)) ;
end proc:

f[p_, e_] := (e + 1 - e/p^2)*p^(2*e); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Feb 09 2023 *)

a(n) = Sum_{d|n} A008683(n/d)*A000005(d)*d^2.
Dirichlet convolution of A034714 and A008683.
Dirichlet g.f.: zeta^2(s-2)/zeta(s).
From Amiram Eldar, Feb 09 2023: (Start)
Multiplicative with a(p^e) = (e + 1 - e/p^2)*p^(2*e).
Sum_{k=1..n} a(k) ~ (log(n) + 2*gamma - 1/3 - zeta'(3)/zeta(3)) * n^3 / (3*zeta(3)), where gamma is Euler's constant (A001620). (End)
From Peter Bala, Jan 16 2024: (Start)
a(n) = Sum_{1 <= i, j <= n} gcd(i, j, n)^2. Cf. A069097.
a(n) = Sum_{d divides n} d^2 * J_2(n/d), where J_2(n) = A007434(n). (End)

A341772 a(n) = Sum_{d|n} phi(d) * J_2(n/d).

Original entry on oeis.org

1, 4, 10, 17, 28, 40, 54, 70, 94, 112, 130, 170, 180, 216, 280, 284, 304, 376, 378, 476, 540, 520, 550, 700, 716, 720, 858, 918, 868, 1120, 990, 1144, 1300, 1216, 1512, 1598, 1404, 1512, 1800, 1960, 1720, 2160, 1890, 2210, 2632, 2200, 2254, 2840, 2682, 2864, 3040, 3060, 2860, 3432, 3640

Offset: 1

Ilya Gutkovskiy, Feb 19 2021

Dirichlet convolution of Euler totient function phi (A000010) with Jordan function J_2 (A007434).

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000010, A000203, A001615, A002618, A007427, A007431, A007434, A008683, A023900, A029935, A064987, A069097, A321322, A322577, A338164.

Jordan2[n_] := Sum[MoebiusMu[n/d] d^2, {d, Divisors[n]}]; a[n_] := Sum[EulerPhi[d] Jordan2[n/d], {d, Divisors[n]}]; Table[a[n], {n, 55}]
f[p_, e_] := p^(e-3)*(p-1)*(p^e*(p+1)^2-p); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, May 31 2024 *)

J2(n) = sumdiv(n, d, d^2 * moebius(n/d)); \\ A007434
a(n) = sumdiv(n, d, eulerphi(d) * J2(n/d)); \\ Michel Marcus, Feb 20 2021

Dirichlet g.f.: zeta(s-1) * zeta(s-2) / zeta(s)^2.
a(n) = Sum_{k=1..n} J_2(gcd(n,k)).
a(n) = Sum_{d|n} psi(d) * phi(d) * phi(n/d).
a(n) = Sum_{d|n} d * phi(d) * A029935(n/d).
a(n) = Sum_{d|n} d * sigma(d) * A007427(n/d).
a(n) = Sum_{d|n} d * A321322(n/d).
a(n) = Sum_{d|n} d * A023900(d) * A338164(n/d).
a(n) = Sum_{d|n} d^2 * A007431(n/d).
a(n) = Sum_{d|n} mu(n/d) * A069097(d).
Sum_{k=1..n} a(k) ~ Pi^2 * n^3 / (18*zeta(3)^2). - Vaclav Kotesovec, Feb 20 2021
a(n) = Sum_{k=1..n} J_2(n/gcd(n,k))*phi(gcd(n,k))/phi(n/gcd(n,k)). - Richard L. Ollerton, May 07 2021
a(n) = Sum_{1 <= i, j <= n}  phi(gcd(i, j, n)). - Peter Bala, Jan 21 2024
Multiplicative with a(p^e) = p^(e-3)*(p-1)*(p^e*(p+1)^2-p). - Amiram Eldar, May 31 2024

A338165 Dirichlet g.f.: (zeta(s-3) / zeta(s))^2.

Original entry on oeis.org

1, 14, 52, 161, 248, 728, 684, 1680, 2080, 3472, 2660, 8372, 4392, 9576, 12896, 16576, 9824, 29120, 13716, 39928, 35568, 37240, 24332, 87360, 46376, 61488, 74412, 110124, 48776, 180544, 59580, 157696, 138320, 137536, 169632, 334880, 101304, 192024, 228384, 416640

Offset: 1

Ilya Gutkovskiy, Oct 14 2020

Dirichlet convolution of Jordan function J_3 (A059376) with itself.

Wikipedia, Jordan's totient function

Cf. A000005, A007427, A029935, A059376, A338164.

Jordan3[n_] := Sum[d^3 MoebiusMu[n/d], {d, Divisors[n]}]; a[n_] := Sum[Jordan3[d] Jordan3[n/d], {d, Divisors[n]}]; Table[a[n], {n, 1, 40}]
a[1] = 1; f[p_, e_] := p^(3 e - 6) (p^6 + e (p^3 - 1)^2 - 1); a[n_] := Times @@ f @@@ FactorInteger[n]; Table[a[n], {n, 1, 40}]

Multiplicative with a(p^e) = p^(3*e - 6) * (p^6 + e * (p^3 - 1)^2 - 1).
a(n) = Sum_{d|n} J_3(d) * J_3(n/d).
a(n) = Sum_{d|n} d^3 * tau(d) * A007427(n/d), where tau = A000005.
(1/tau(n)) * Sum_{d|n} a(d) * tau(n/d) = n^3.
Sum_{k=1..n} a(k) ~ 2025 * n^4 * ((log(n) + 2*gamma - 1/4)/Pi^8 - 180*zeta'(4) / Pi^12), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Oct 14 2020

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A360428 Inverse Mobius transformation of A338164.

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A341772 a(n) = Sum_{d|n} phi(d) * J_2(n/d).

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A338165 Dirichlet g.f.: (zeta(s-3) / zeta(s))^2.

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