A069097 -id:A069097 - cp's OEIS frontend

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

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

A372927 a(n) = Sum_{1 <= x_1, x_2 <= n} gcd(x_1, x_2, n)^5.

Original entry on oeis.org

1, 35, 251, 1132, 3149, 8785, 16855, 36272, 61065, 110215, 161171, 284132, 371461, 589925, 790399, 1160896, 1420145, 2137275, 2476459, 3564668, 4230605, 5640985, 6436871, 9104272, 9841225, 13001135, 14839443, 19079860, 20511989, 27663965, 28630111, 37149440, 40453921

Offset: 1

Seiichi Manyama, May 17 2024

Seiichi Manyama, Table of n, a(n) for n = 1..10000
Peter Bala, GCD sum theorems. Two Multivariable Cesaro Type Identities.

Cf. A069097, A360428, A368743, A372926.
Cf. A001158, A008683.
Cf. A013662, A013664, A088246.

f[p_, e_] := p^(2*e-2) * (p^2 * (p^(3*e+3)-1) - p^(3*e) + 1)/(p^3-1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, May 21 2024 *)

a(n) = sumdiv(n, d, moebius(n/d)*d^2*sigma(d, 3));

a(n) = Sum_{1 <= x_1, x_2, x_3, x_4, x_5 <= n} gcd(x_1, x_2, x_3, x_4, x_5, n)^2.
a(n) = Sum_{d|n} mu(n/d) * d^2 * sigma_3(d), where mu is the Moebius function A008683.
From Amiram Eldar, May 21 2024: (Start)
Multiplicative with a(p^e) = p^(2*e-2) * (p^2 * (p^(3*e+3)-1) - p^(3*e) + 1)/(p^3-1).
Dirichlet g.f.: zeta(s-2)*zeta(s-5)/zeta(s).
Sum_{k=1..n} a(k) ~ c * n^6 / 6, where c = zeta(4)/zeta(6) = 21/(2*Pi^2) = 1.0638724... (A088246). (End)

A143309 Triangle read by rows, A054525 * A143308, 1<=k<=n.

Original entry on oeis.org

1, 3, 2, 5, 3, 3, 8, 6, 4, 4, 9, 5, 5, 5, 5, 15, 13, 9, 6, 6, 6, 13, 7, 7, 7, 7, 7, 7, 20, 16, 12, 12, 8, 8, 8, 8, 21, 15, 15, 9, 9, 9, 9, 9, 9, 27, 23, 15, 15, 15, 10, 10, 10, 10, 10, 21, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 40, 36, 32, 26, 18, 18, 12, 12, 12, 12, 12, 12, 25, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13

Offset: 1

Gary W. Adamson, Aug 06 2008

Row sums = A069097: (1, 5, 11, 22, 29, 55,...).

Left border = A018804: (1, 3, 5, 8, 9, 15,...).

			First few rows of the triangle =
1;
3, 2;
5, 3, 3;
8, 6, 4, 4;
9, 5, 5, 5, 5;
15, 13, 9, 6, 6, 6;
13, 7, 7, 7, 7, 7, 7;
...

Cf. A018804, A069097, A143308.

Mobius transform of triangle A143308.

a(56) = 10 removed and more terms from Georg Fischer, Jul 13 2023

A143312 Triangle read by rows, A054525 * A143311, 1<=k<=n.

Original entry on oeis.org

1, 3, 2, 8, 0, 3, 12, 6, 0, 4, 24, 0, 0, 0, 5, 24, 16, 9, 0, 0, 6, 48, 0, 0, 0, 0, 0, 7, 48, 24, 0, 12, 0, 0, 0, 8, 72, 0, 24, 0, 0, 0, 0, 0, 9, 72, 48, 0, 0, 15, 0, 0, 0, 0, 10, 120, 0, 0, 0, 0, 0, 0, 0, 0, 0, 11, 96, 48, 36, 32, 0, 18, 0, 0, 0, 0, 0, 12, 168, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 13

Offset: 1

Gary W. Adamson, Aug 06 2008

Left border = A007434.

Row sums = A069097: (1, 5, 11, 2, 29, 55,...)

			First few rows of the triangle =
1;
3, 2;
8, 0, 3;
12, 6, 0, 4;
24, 0, 0, 0, 5;
24, 16, 9, 0, 0, 6;
48, 0, 0, 0, 0, 0, 7;
48, 24, 0, 12, 0, 0, 0, 8;
...

Cf. A054525, A143311, A007434, A069097.

Mobius transform of triangle A143311

A304275 a(n) = Sum_{k = 1..n} gcd(k,n) / cos(Pi*k/n)^2 for odd n.

Original entry on oeis.org

1, 11, 29, 55, 105, 131, 181, 319, 305, 379, 605, 551, 745, 963, 869, 991, 1441, 1595, 1405, 1991, 1721, 1891, 3045, 2255, 2737, 3355, 2861, 3799, 4169, 3539, 3781, 5775, 5249, 4555, 6061, 5111, 5401, 8195, 7205, 6319, 8721, 6971, 8845

Offset: 1

Hugo Pfoertner, May 10 2018

Hugo Pfoertner, Table of n, a(n) for n = 1..1000
Nikolai Osipov (Proposer), Problem 12003, Amer. Math. Monthly 124 (No. 8, Oct. 2017), page 754.
Nikolay Osipov, Proposer's solution to Problem 12003

Cf. A002117, A069097.

seq( round( add(igcd(k, 2*n+1)/cos(Pi*k/(2*n+1))^2, k = 1..2*n+1) ), n = 0..40); # Peter Bala, Dec 26 2023

f[p_, e_] := p^(e-1)*(p^e*(p+1)-1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[2*n - 1]; Array[a, 50] (* Amiram Eldar, Dec 28 2023 *)

a(n) = n = 2*n-1; round(sum(k=1, n, gcd(k,n) / cos(Pi*k/n)^2)); \\ Michel Marcus, May 10 2018

a(n) = A069097(2*n-1). - Peter Bala, Dec 26 2023
a(n) = (1/3)*Sum_{k = 1..4*n-2} (-1)^k*gcd(k,4*n-2)^2. - Conjectured by Peter Bala, Dec 26 2023; proved by Nikolay Osipov, Oct 05 2024
Sum_{k=1..n} a(k) ~ c * n^3, where c = 4*Pi^2 / (21*zeta(3)) = 1.563923... . - Amiram Eldar, Dec 28 2023

A351654 Dirichlet g.f.: zeta(s) / (zeta(s-1) * zeta(s-2)).

Original entry on oeis.org

1, -5, -11, 3, -29, 55, -55, 3, 16, 145, -131, -33, -181, 275, 319, 3, -305, -80, -379, -87, 605, 655, -551, -33, 96, 905, 16, -165, -869, -1595, -991, 3, 1441, 1525, 1595, 48, -1405, 1895, 1991, -87, -1721, -3025, -1891, -393, -464, 2755, -2255, -33, 288, -480, 3355, -543, -2861, -80, 3799

Offset: 1

Ilya Gutkovskiy, Feb 16 2022

Dirichlet inverse of A069097.

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

Cf. A000005, A023900, A046970, A055615, A069097, A101035, A328254, A334657.

A069097[n_] := Sum[GCD[n, k]^2, {k, 1, n}]; a[1] = 1; a[n_] := a[n] = -Sum[A069097[n/d] a[d], {d, Most @ Divisors[n]}]; Table[a[n], {n, 1, 55}]
f[p_, e_] := If[e == 1, 0, p^3] - p^2 - p + 1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, May 25 2025 *)

up_to = 20000;
DirInverseCorrect(v) = { my(u=vector(#v)); u[1] = (1/v[1]); for(n=2, #v, u[n] = (-u[1]*sumdiv(n, d, if(dA069097(n) = sumdiv(n, d, moebius(n/d)*d*sigma(d));
v351654 = DirInverseCorrect(vector(up_to, n, A069097(n)));
A351654(n) = v351654[n]; \\ Antti Karttunen, Feb 16 2022

for(n=1, 100, print1(direuler(p=2, n, (1 - p*X)*(1 - p^2*X)/(1 - X))[n], ", ")) \\ Vaclav Kotesovec, Feb 16 2022

a(1) = 1; a(n) = -Sum_{d|n, d < n} A069097(n/d) * a(d).
a(n) = Sum_{d|n} A023900(n/d) * A334657(d).
a(n) = Sum_{d|n} A046970(n/d) * A055615(d).
a(n) = Sum_{d|n} A000005(n/d) * A328254(d).
Multiplicative with a(p) = -p^2 - p + 1, and a(p^e) = p^3 - p^2 - p + 1 for e >= 2. - Amiram Eldar, May 25 2025

A372918 a(n) = Sum_{k=1..n} gcd(k^3,n^2).

Original entry on oeis.org

1, 5, 11, 26, 29, 55, 55, 148, 141, 145, 131, 286, 181, 275, 319, 680, 305, 705, 379, 754, 605, 655, 551, 1628, 1145, 905, 2367, 1430, 869, 1595, 991, 3408, 1441, 1525, 1595, 3666, 1405, 1895, 1991, 4292, 1721, 3025, 1891, 3406, 4089, 2755, 2255, 7480, 4501, 5725, 3355

Offset: 1

Seiichi Manyama, May 16 2024

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

Cf. A069097, A078430, A343497, A372882.

a[n_] := Sum[GCD[k^3, n^2], {k, 1, n}]; Array[a, 100] (* Amiram Eldar, May 24 2024 *)

PARI
```
a(n) = sum(k=1, n, gcd(k^3, n^2));
```

Multiplicative with a(p^e) = p^e * (1 + ((p-1)/p) * Sum_{i=1..2*e} p^(floor(2*i/3))). - Amiram Eldar, May 24 2024

A332778 a(1) = 1; a(n) = Sum_{d|n, d < n} phi(n/d) * a(d)^2.

Original entry on oeis.org

1, 1, 2, 3, 4, 8, 6, 15, 14, 24, 10, 96, 12, 48, 56, 255, 16, 344, 18, 656, 108, 120, 22, 9840, 84, 168, 434, 2448, 28, 4608, 30, 65535, 260, 288, 264, 137376, 36, 360, 360, 432512, 40, 16776, 42, 14720, 7208, 528, 46, 96974880, 258, 9464, 608, 28656, 52, 425864, 600

Offset: 1

Ilya Gutkovskiy, Feb 23 2020

nonn

Cf. A000010, A006874, A007018, A023900, A069097, A082588.

a[1] = 1; a[n_] := Sum[If[d < n, EulerPhi[n/d] a[d]^2, 0], {d, Divisors[n]}]; Table[a[n], {n, 1, 55}]
a[1] = 1; a[n_] := Sum[a[GCD[n, k]]^2, {k, 1, n - 1}]; Table[a[n], {n, 1, 55}]

a(1) = 1; a(n) = Sum_{k=1..n-1} a(gcd(n, k))^2.

A348011 a(n) = phi(n^2) * Sum_{d|n} 2^omega(d) / d.

Original entry on oeis.org

1, 4, 10, 20, 28, 40, 54, 88, 102, 112, 130, 200, 180, 216, 280, 368, 304, 408, 378, 560, 540, 520, 550, 880, 740, 720, 954, 1080, 868, 1120, 990, 1504, 1300, 1216, 1512, 2040, 1404, 1512, 1800, 2464, 1720, 2160, 1890, 2600, 2856, 2200, 2254, 3680, 2730, 2960

Offset: 1

Ilya Gutkovskiy, Sep 24 2021

Cf. A000010, A001221, A002618, A007434, A034444, A047994, A060648, A062355, A069097, A191414, A341772.

Table[EulerPhi[n^2] DivisorSum[n, 2^PrimeNu[#]/# &], {n, 50}]
f[p_, e_] := p^(e - 1) ((p + 1) p^e - 2); a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 50]

a(n) = eulerphi(n^2)*sumdiv(n, d, 2^omega(d)/d); \\ Michel Marcus, Sep 24 2021

Multiplicative with a(p^e) = p^(e-1) * ((p + 1) * p^e - 2).
a(n) = Sum_{k=1..n, gcd(n,k) = 1} gcd(n,k-1)^2.
a(n) = Sum_{k=1..n} uphi(gcd(n,k)^2).
a(n) = Sum_{d|n} phi(n/d) * uphi(d^2).
Sum_{k=1..n} a(k) ~ c * n^3, where c = (Pi^2/18) * Product_{p prime} (1 - 2/p^3 + 1/p^4) = 0.4083249979... . - Amiram Eldar, Nov 05 2022

cp's OEIS Frontend

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

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A372927 a(n) = Sum_{1 <= x_1, x_2 <= n} gcd(x_1, x_2, n)^5.

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A143309 Triangle read by rows, A054525 * A143308, 1<=k<=n.

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A143312 Triangle read by rows, A054525 * A143311, 1<=k<=n.

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A304275 a(n) = Sum_{k = 1..n} gcd(k,n) / cos(Pi*k/n)^2 for odd n.

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A351654 Dirichlet g.f.: zeta(s) / (zeta(s-1) * zeta(s-2)).

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A372918 a(n) = Sum_{k=1..n} gcd(k^3,n^2).

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A332778 a(1) = 1; a(n) = Sum_{d|n, d < n} phi(n/d) * a(d)^2.

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