A029935 -id:A029935 - cp's OEIS frontend

A342534 a(n) = Sum_{k=1..n} phi(gcd(k, n))^2.

1, 2, 6, 7, 20, 12, 42, 26, 50, 40, 110, 42, 156, 84, 120, 100, 272, 100, 342, 140, 252, 220, 506, 156, 484, 312, 438, 294, 812, 240, 930, 392, 660, 544, 840, 350, 1332, 684, 936, 520, 1640, 504, 1806, 770, 1000, 1012, 2162, 600, 2022, 968, 1632, 1092, 2756, 876, 2200, 1092, 2052

Offset: 1

Seiichi Manyama, Mar 15 2021

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

Cf. A000010, A029935, A029939, A065464, A306633, A338997, A342535.

a[n_] := DivisorSum[n, EulerPhi[n/#] * EulerPhi[#]^2 &]; Array[a, 50] (* Amiram Eldar, Mar 15 2021 *)

a(n) = sum(k=1, n, eulerphi(gcd(k, n))^2);

a(n) = sumdiv(n, d, eulerphi(n/d)*eulerphi(d)^2);

a(n) = Sum_{d|n} phi(n/d) * phi(d)^2.
a(n) = Sum_{k=1..n} phi(gcd(k,n))*phi(n/gcd(k,n)). - Richard L. Ollerton, May 10 2021
From Amiram Eldar, Nov 15 2022: (Start)
Multiplicative with a(p^e) = (p-1)*(p^(e-2) - p^(2*e-3) + p^(2*e-1)).
Sum_{k=1..n} a(k) ~ c * n^3, where c = zeta(2)/(3*zeta(3)) * Product_{p prime} (1 - (2*p-1)/p^3) = A306633 * A065464 / 3 = 0.195343... . (End)

A342535 a(n) = Sum_{k=1..n} phi(gcd(k, n))^3.

Original entry on oeis.org

1, 2, 10, 11, 68, 20, 222, 78, 238, 136, 1010, 110, 1740, 444, 680, 604, 4112, 476, 5850, 748, 2220, 2020, 10670, 780, 8276, 3480, 6330, 2442, 21980, 1360, 27030, 4792, 10100, 8224, 15096, 2618, 46692, 11700, 17400, 5304, 64040, 4440, 74130, 11110, 16184, 21340, 97382, 6040

Offset: 1

Seiichi Manyama, Mar 15 2021

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

Cf. A000010, A002117, A029935, A342470, A342534.

a[n_] := DivisorSum[n, EulerPhi[n/#] * EulerPhi[#]^3 &]; Array[a, 50] (* Amiram Eldar, Mar 15 2021 *)
Table[Sum[EulerPhi[GCD[k,n]]^3,{k,n}],{n,50}] (* Harvey P. Dale, Jul 15 2021 *)

a(n) = sum(k=1, n, eulerphi(gcd(k, n))^3);

a(n) = sumdiv(n, d, eulerphi(n/d)*eulerphi(d)^3);

a(n) = Sum_{d|n} phi(n/d) * phi(d)^3.
a(n) = Sum_{k=1..n} phi(gcd(k,n))*phi(n/gcd(k,n))^2. - Richard L. Ollerton, May 10 2021
Sum_{k=1..n} a(k) ~ c * n^3, where c = (zeta(3)/4) * Product_{p prime} (1 - 3/p^2 + 3/p^3 - 2/p^4 + 3/p^6 - 3/p^7 + 1/p^8) = 0.093622450005... . - Amiram Eldar, Nov 15 2022

A127192 Triangle read by rows: square of A054523.

Original entry on oeis.org

1, 2, 1, 4, 0, 1, 5, 2, 0, 1, 8, 0, 0, 0, 1, 8, 4, 2, 0, 0, 1, 12, 0, 0, 0, 0, 0, 1, 12, 5, 0, 2, 0, 0, 0, 1, 16, 0, 4, 0, 0, 0, 0, 0, 1, 16, 8, 0, 0, 2, 0, 0, 0, 0, 1, 20, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 20, 8, 5, 4, 0, 2, 0, 0, 0, 0, 0, 1, 24, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1

Offset: 1

Gary W. Adamson, Jan 07 2007

Row sums = A018804: (1, 3, 5, 8, 9, 15, ...), sum of gcd(k,n) for 1<= k <= n. Left column = A029935: (1, 2, 4, 5, 8, 8, 12, 12, ...). A127192 * d(n) = d(n) * n, or A127192 * A000005 = A038040 = (1, 4, 6, 12, 10, 24, 14, ...).

Column k contains terms of A029935 interspersed with (k-1) zeros.

			First few rows of the triangle:
   1;
   2, 1;
   4, 0, 1;
   5, 2, 0, 1;
   8, 0, 0, 0, 1;
   8, 4, 2, 0, 0, 1;
  12, 0, 0, 0, 0, 0, 1;
  12, 5, 0, 2, 0, 0, 0, 1;
  16, 0, 4, 0, 0, 0, 0, 0, 1;
  ...

Cf. A000005, A018804, A029935, A038040, A054523.

a(20) ff. corrected and more terms from Georg Fischer, May 31 2023

A159937 Triangle read by rows, A054525 * A127478, as infinite lower triangular matrices.

Original entry on oeis.org

1, 1, 1, 2, 0, 2, 2, 1, 0, 2, 4, 0, 0, 0, 4, 2, 2, 2, 0, 0, 2, 6, 0, 0, 0, 0, 0, 6, 4, 2, 0, 2, 0, 0, 0, 4, 6, 0, 4, 0, 0, 0, 0, 0, 6, 4, 4, 0, 0, 4, 0, 0, 0, 0, 4, 10, 0, 0, 0, 0, 0, 0, 0, 0, 0, 10, 4, 2, 4, 4, 0, 2, 0, 0, 0, 0, 0, 4

Offset: 1

Gary W. Adamson, Apr 26 2009

Row sums = A029935: (1, 2, 4, 5, 8, 8,...). Right and left borders = A000010, phi(n).

			First few rows of the triangle =
1;
1, 1;
2, 0, 2;
2, 1, 0, 2;
4, 0, 0, 0, 4;
2, 2, 2, 0, 0, 2;
6, 0, 0, 0, 0, 0, 6;
4, 2, 0, 2, 0, 0, 0, 4;
6, 0, 4, 0, 0, 0, 0, 0, 6;
4, 4, 0, 0, 4, 0, 0, 0, 0, 4;
10, 0, 0, 0, 0, 0, 0, 0, 0, 0, 10;
4, 2, 4, 4, 0, 2, 0, 0, 0, 0, 0, 4;
12, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 12;
6, 6, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 0, 6;
8, 0, 8, 0, 8, 0, 0, 0, 0, 0, 0, 0, 0, 0, 8;
...

Cf. A127478, A029935

A054525 * A127478 = Mobius transform of triangle A127478.

A332686 a(n) = Sum_{k=1..n} phi(k/gcd(n, k)).

Original entry on oeis.org

1, 2, 3, 5, 7, 9, 13, 18, 21, 23, 33, 33, 47, 49, 51, 67, 81, 76, 103, 97, 103, 119, 151, 135, 163, 173, 185, 189, 243, 185, 279, 280, 265, 299, 291, 291, 397, 379, 369, 371, 491, 381, 543, 491, 455, 553, 651, 539, 653, 610, 643, 683, 831, 689, 743, 753, 801, 887, 1029

Offset: 1

Ilya Gutkovskiy, Feb 19 2020

nonn

Inverse Moebius transform of A053570.

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

Cf. A000010, A002088, A029935, A029939, A053570, A332685.

Table[Sum[EulerPhi[k/GCD[n, k]], {k, 1, n}], {n, 1, 59}]

a(n) = sum(k=1, n, eulerphi(k/gcd(n, k))); \\ Michel Marcus, Feb 21 2020

a(n) = Sum_{k=1..n} phi(lcm(n, k)/n).

a(n) = Sum_{d|n} A053570(d).

A333557 a(n) = Sum_{d|n, gcd(d, n/d) = 1} uphi(d) * uphi(n/d), where uphi = unitary totient function (A047994).

Original entry on oeis.org

1, 2, 4, 6, 8, 8, 12, 14, 16, 16, 20, 24, 24, 24, 32, 30, 32, 32, 36, 48, 48, 40, 44, 56, 48, 48, 52, 72, 56, 64, 60, 62, 80, 64, 96, 96, 72, 72, 96, 112, 80, 96, 84, 120, 128, 88, 92, 120, 96, 96, 128, 144, 104, 104, 160, 168, 144, 112, 116, 192, 120, 120, 192, 126, 192, 160, 132, 192, 176, 192

Offset: 1

Ilya Gutkovskiy, Mar 26 2020

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

Cf. A001221, A029935, A034444, A047994, A055653, A062355, A145388.

uphi[1] = 1; uphi[n_] := Times @@ (#[[1]]^#[[2]] - 1 & /@ FactorInteger[n]); a[n_] := Sum[If[GCD[d, n/d] == 1, uphi[d] uphi[n/d], 0], {d, Divisors[n]}]; Table[a[n], {n, 1, 70}]
Table[Sum[If[GCD[d, n/d] == 1, (-2)^PrimeNu[n/d] 2^PrimeNu[d] d, 0], {d, Divisors[n]}], {n, 1, 70}]
f[p_, e_] := 2*(p^e-1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Apr 30 2023 *)

a(n) = sumdiv(n, d, if (gcd(d, n/d) == 1, (-2)^omega(n/d)*2^omega(d)*d)); \\ Michel Marcus, Mar 27 2020

If n = Product (p_j^k_j) then a(n) = Product (2 * (p_j^k_j - 1)).
a(n) = 2^omega(n) * uphi(n).
a(n) = Sum_{d|n, gcd(d, n/d) = 1} (-2)^omega(n/d) * 2^omega(d) * d.
a(n) = Sum_{d|n, gcd(d, n/d) = 1} (-1)^omega(n/d) * A145388(d).

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

A344683 Dirichlet convolution of the Euler totient function with itself, applied twice.

Original entry on oeis.org

1, 3, 6, 9, 12, 18, 18, 25, 30, 36, 30, 54, 36, 54, 72, 66, 48, 90, 54, 108, 108, 90, 66, 150, 108, 108, 134, 162, 84, 216, 90, 168, 180, 144, 216, 270, 108, 162, 216, 300, 120, 324, 126, 270, 360, 198, 138, 396, 234, 324, 288, 324, 156, 402, 360, 450, 324

Offset: 1

Sebastian Karlsson, Aug 17 2021

Dirichlet convolution of A000010 with A029935.

Sebastian Karlsson, Table of n, a(n) for n = 1..10000

Cf. A000010, A029935, A166633.

f[p_, e_] := (1/2)*(p - 1)*p^(e - 3)*(e^2*(p - 1)^2 + 3*e*(p^2 - 1) + 2*(p^2 + p + 1)); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Aug 17 2021 *)

from sympy import divisors as div, totient as phi
def D(f, g, n):
    return sum(f(d)*g(n//d) for d in div(n))
def phi_o_phi(n):
    return D(phi, phi, n)
def a(n):
    return D(phi, phi_o_phi, n)

Dirichlet g.f.: zeta(s - 1)^3 / zeta(s)^3.
Multiplicative with a(p^e) = (1/2)*(p-1)*p^(e-3)*(e^2*(p-1)^2 + 3*e*(p^2-1) + 2*(p^2 + p + 1)).
Sum_{k=1..n} a(k) ~ 27*n^2/Pi^10 * (2*Pi^4*log(n)^2 - 2*Pi^4*log(n)*(1 + 6*log(2) - 72*(1/12 - zeta'(-1)) + 6*log(Pi)) + Pi^4*(1 + 6*gamma*(2*gamma - 1) - 12*sg1) + 864*zeta'(2)^2 - 36*Pi^2*((6*gamma - 1)*zeta'(2) + zeta''(2))), where gamma is the Euler-Mascheroni constant A001620 and sg1 is the first Stieltjes constant (see A082633). - Vaclav Kotesovec, Jun 24 2022

A063754 Dirichlet convolution of totient and cototient.

Original entry on oeis.org

0, 1, 1, 3, 1, 7, 1, 8, 5, 11, 1, 20, 1, 15, 13, 20, 1, 31, 1, 32, 17, 23, 1, 52, 9, 27, 21, 44, 1, 71, 1, 48, 25, 35, 21, 88, 1, 39, 29, 84, 1, 99, 1, 68, 61, 47, 1, 128, 13, 83, 37, 80, 1, 123, 29, 116, 41, 59, 1, 200, 1, 63, 81, 112, 33, 155, 1, 104, 49, 159, 1, 228, 1, 75, 101

Offset: 1

Labos Elemer, Aug 14 2001

a(n) = 1 if and only if n is prime. - Robert Israel, Feb 04 2018

a(n) = n+1 if and only if n = 2*p with p an odd prime (A100484 \ {4}). - Bernard Schott, Jun 19 2023

			n = 24: divisors = {1, 2, 3, 4, 6, 8, 12, 24}, d-phi(d) = {0, 1, 1, 2, 4, 4, 8, 16}, phi(n/d) = {8, 4, 4, 2, 2, 2, 1, 1}, products = {0, 4, 4, 4, 8, 8, 8, 16}, a(24) = 52.

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

Cf. A000010, A018804, A029935, A051953, A100484.

Cf. A001620, A013661, A306016.

f:= n -> add(numtheory:-phi(d)*(n/d - numtheory:-phi(n/d)), d=numtheory:-divisors(n)):
map(f, [$1..100]); # Robert Israel, Feb 04 2018

f1[p_, e_] := (e*(p - 1)/p + 1)*p^e; f2[p_, e_] := (e+1)*(p^e - p^(e-1)) - (e-1)*(p^(e-1) - p^(e-2)); a[n_] := Times @@ f1 @@@ (fct = FactorInteger[n]) - Times @@ f2 @@@ fct; a[1] = 0; Array[a, 100] (* Amiram Eldar, Apr 28 2023 *)

a(n) = sumdiv(n, d, eulerphi(d)*(n/d - eulerphi(n/d))); \\ Michel Marcus, Feb 05 2018

a(n) = Sum_{d|n} A000010(d)*A051953(n/d).
From Richard L. Ollerton, May 06 2021: (Start)
a(n) = Sum_{k=1..n} A051953(gcd(n,k)).
a(n) = Sum_{k=1..n} A051953(n/gcd(n,k))*A000010(gcd(n,k))/A000010(n/gcd(n,k)).
a(n) = A018804(n) - A029935(n). (End)
Sum_{k=1..n} a(k) ~ (1/(2*zeta(2)))*(1 - 1/zeta(2)) * n^2 * (log(n) + 2*gamma - 1/2 - ((zeta(2)-2)/(zeta(2)-1))*(zeta'(2)/zeta(2))), where gamma is Euler's constant (A001620). - Amiram Eldar, Jan 11 2024

Offset corrected by Robert Israel, Feb 04 2018

A127627 Triangle T(n,k) = A054525(n,k)*A018804(k), read by rows 1<=k<=n.

Original entry on oeis.org

1, -1, 3, -1, 0, 5, 0, -3, 0, 8, -1, 0, 0, 0, 9, 1, -3, -5, 0, 0, 15, -1, 0, 0, 0, 0, 0, 13, 0, 0, 0, -8, 0, 0, 0, 20, 0, 0, -5, 0, 0, 0, 0, 0, 21, 1, -3, 0, 0, -9, 0, 0, 0, 0, 27, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 21, 0, 3, 0, -8, 0, -15, 0, 0, 0, 0, 0, 40

Offset: 1

Gary W. Adamson, Jan 20 2007

			First few rows of the triangle are:
1;
-1, 3;
-1, 0, 5;
0, -3, 0, 8;
-1, 0, 0, 0, 9;
1, -3,-5, 0, 0, 15;
...

Cf. A054525, A018804, A029935 (row sums).

A018804 := proc(n)
    sum(igcd(n, j), j=1..n):
end proc:
A127627 := proc(n,k)
    A054525(n,k)*A018804(k) ;
end proc: # R. J. Mathar, Oct 21 2012

T(n,1) = A008683(n).

T(n,n) = A018804(n).

cp's OEIS Frontend

A342534 a(n) = Sum_{k=1..n} phi(gcd(k, n))^2.

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A342535 a(n) = Sum_{k=1..n} phi(gcd(k, n))^3.

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A127192 Triangle read by rows: square of A054523.

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A159937 Triangle read by rows, A054525 * A127478, as infinite lower triangular matrices.

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A332686 a(n) = Sum_{k=1..n} phi(k/gcd(n, k)).

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A333557 a(n) = Sum_{d|n, gcd(d, n/d) = 1} uphi(d) * uphi(n/d), where uphi = unitary totient function (A047994).

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

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A344683 Dirichlet convolution of the Euler totient function with itself, applied twice.

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