A332517 -id:A332517 - cp's OEIS frontend

A343497 a(n) = Sum_{k=1..n} gcd(k, n)^3.

1, 9, 29, 74, 129, 261, 349, 596, 789, 1161, 1341, 2146, 2209, 3141, 3741, 4776, 4929, 7101, 6877, 9546, 10121, 12069, 12189, 17284, 16145, 19881, 21321, 25826, 24417, 33669, 29821, 38224, 38889, 44361, 45021, 58386, 50689, 61893, 64061, 76884, 68961, 91089, 79549, 99234, 101781

Offset: 1

Seiichi Manyama, Apr 17 2021

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

Column 3 of A343510.

Cf. A000010, A001157 (sigma_2(n)), A018804, A054610, A069097, A309323, A332517, A342423, A342433, A343498, A343499, A343513.

A343497:= func< n | (&+[d^3*EulerPhi(Floor(n/d)): d in Divisors(n)]) >;
[A343497(n): n in [1..50]]; // G. C. Greubel, Jun 24 2024

with(numtheory):
seq(add(phi(n/d) * d^3, d in divisors(n)), n = 1..50); # Peter Bala, Jan 20 2024

a[n_] := Sum[GCD[k, n]^3, {k, 1, n}]; Array[a, 50] (* Amiram Eldar, Apr 18 2021 *)
f[p_, e_] := p^(e - 1)*((p^2 + p + 1)*p^(2*e) - 1)/(p + 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 50] (* Amiram Eldar, Nov 22 2022 *)
A343497[n_]:= DivisorSum[n, #^3*EulerPhi[n/#] &]; Table[A343497[n], {n, 50}] (* G. C. Greubel, Jun 24 2024 *)

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

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

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

my(N=40, x='x+O('x^N)); Vec(sum(k=1, N, eulerphi(k)*x^k*(1+4*x^k+x^(2*k))/(1-x^k)^4))

def A343497(n): return sum(k^3*euler_phi(n/k) for k in (1..n) if (k).divides(n))
[A343497(n) for n in range(1,51)] # G. C. Greubel, Jun 24 2024

a(n) = Sum_{d|n} phi(n/d) * d^3.
a(n) = Sum_{d|n} mu(n/d) * d * sigma_2(d).
G.f.: Sum_{k >= 1} phi(k) * x^k * (1 + 4*x^k + x^(2*k))/(1 - x^k)^4.
Dirichlet g.f.: zeta(s-1) * zeta(s-3) / zeta(s). - Ilya Gutkovskiy, Apr 18 2021
Sum_{k=1..n} a(k) ~ 45*zeta(3)*n^4 / (2*Pi^4). - Vaclav Kotesovec, May 20 2021
Multiplicative with a(p^e) = p^(e-1)*((p^2+p+1)*p^(2*e) - 1)/(p+1). - Amiram Eldar, Nov 22 2022
a(n) = Sum_{1 <= i, j, k <= n} gcd(i, j, k, n) = Sum_{d divides n} d * J_3(n/d), where the Jordan totient function J_3(n) = A059376(n). - Peter Bala, Jan 20 2024

A343498 a(n) = Sum_{k=1..n} gcd(k, n)^4.

Original entry on oeis.org

1, 17, 83, 274, 629, 1411, 2407, 4388, 6729, 10693, 14651, 22742, 28573, 40919, 52207, 70216, 83537, 114393, 130339, 172346, 199781, 249067, 279863, 364204, 393145, 485741, 545067, 659518, 707309, 887519, 923551, 1123472, 1216033, 1420129, 1514003, 1843746, 1874197

Offset: 1

Seiichi Manyama, Apr 17 2021

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

Column 4 of A343510.

Cf. A000010, A001158 (sigma_3(n)), A018804, A054611, A069097, A332517, A342423, A342433, A343497, A343499, A343514.

A343498:= func< n | (&+[d^4*EulerPhi(Floor(n/d)): d in Divisors(n)]) >;
[A343498(n): n in [1..50]]; // G. C. Greubel, Jun 24 2024

a[n_] := Sum[GCD[k, n]^4, {k, 1, n}]; Array[a, 50] (* Amiram Eldar, Apr 18 2021 *)
f[p_, e_] := p^(e-1)*(p^(3*e+4) - p^(3*e) - p + 1)/(p^3-1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 50] (* Amiram Eldar, Nov 22 2022 *)

PARI
```
a(n) = sum(k=1, n, gcd(k, n)^4);
```

a(n) = sumdiv(n, d, eulerphi(n/d)*d^4);

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

my(N=40, x='x+O('x^N)); Vec(sum(k=1, N, eulerphi(k)*x^k*(1+11*x^k+11*x^(2*k)+x^(3*k))/(1-x^k)^5))

def A343498(n): return sum(k^4*euler_phi(n/k) for k in (1..n) if (k).divides(n))
[A343498(n) for n in range(1,51)] # G. C. Greubel, Jun 24 2024

a(n) = Sum_{d|n} phi(n/d) * d^4.
a(n) = Sum_{d|n} mu(n/d) * d * sigma_3(d).
G.f.: Sum_{k >= 1} phi(k) * x^k * (1 + 11*x^k + 11*x^(2*k) + x^(3*k))/(1 - x^k)^5.
Dirichlet g.f.: zeta(s-1) * zeta(s-4) / zeta(s). - Ilya Gutkovskiy, Apr 18 2021
Sum_{k=1..n} a(k) ~ Pi^4 * n^5 / (450*zeta(5)). - Vaclav Kotesovec, May 20 2021
Multiplicative with a(p^e) = p^(e-1)*(p^(3*e+4) - p^(3*e) - p + 1)/(p^3-1). - Amiram Eldar, Nov 22 2022
a(n) = Sum_{1 <= i, j, k, l <= n} gcd(i, j, k, l, n) = Sum_{d divides n} d * J_4(n/d), where the Jordan totient function J_4(n) = A059377(n). - Peter Bala, Jan 18 2024

A343510 Square array T(n,k), n >= 1, k >= 1, read by antidiagonals, where T(n,k) = Sum_{j=1..n} gcd(j, n)^k.

Original entry on oeis.org

1, 1, 3, 1, 5, 5, 1, 9, 11, 8, 1, 17, 29, 22, 9, 1, 33, 83, 74, 29, 15, 1, 65, 245, 274, 129, 55, 13, 1, 129, 731, 1058, 629, 261, 55, 20, 1, 257, 2189, 4162, 3129, 1411, 349, 92, 21, 1, 513, 6563, 16514, 15629, 8085, 2407, 596, 105, 27, 1, 1025, 19685, 65794, 78129, 47515, 16813, 4388, 789, 145, 21

Offset: 1

Seiichi Manyama, Apr 17 2021

			G.f. of column 3: Sum_{i>=1} phi(i) * (x^i + 4*x^(2*i) + x^(3*i))/(1 - x^i)^4.
Square array begins:
   1,  1,   1,    1,     1,      1,      1, ...
   3,  5,   9,   17,    33,     65,    129, ...
   5, 11,  29,   83,   245,    731,   2189, ...
   8, 22,  74,  274,  1058,   4162,  16514, ...
   9, 29, 129,  629,  3129,  15629,  78129, ...
  15, 55, 261, 1411,  8085,  47515, 282381, ...
  13, 55, 349, 2407, 16813, 117655, 823549, ...

Seiichi Manyama, Antidiagonals n = 1..140, flattened
Peter Bala, GCD sum theorems. Two Multivariable Cesaro Type Identities.
Eric Weisstein's World of Mathematics, Eulerian Number and Euler's Number Triangle

Columns k=1..7 give A018804, A069097, A343497, A343498, A343499, A343508, A343509.
T(n-2,n) gives A342432.
T(n-1,n) gives A342433.
T(n,n) gives A332517.
T(n,n+1) gives A321294.
Cf. A008292, A343516.

T[n_, k_] := DivisorSum[n, EulerPhi[n/#] * #^k &]; Table[T[k, n - k + 1], {n, 1, 11}, {k, 1, n}] // Flatten (* Amiram Eldar, Apr 18 2021 *)

PARI
```
T(n, k) = sum(j=1, n, gcd(j, n)^k);
```

T(n, k) = sumdiv(n, d, eulerphi(n/d)*d^k);

T(n, k) = sumdiv(n, d, moebius(n/d)*d*sigma(d, k-1));

G.f. of column k: Sum_{i>=1} phi(i) * ( Sum_{j=1..k} A008292(k, j) * x^(i*j) )/(1 - x^i)^(k+1).
T(n,k) = Sum_{d|n} phi(n/d) * d^k.
T(n,k) = Sum_{d|n} mu(n/d) * d * sigma_{k-1}(d).
Dirichlet g.f. of column k: zeta(s-1) * zeta(s-k) / zeta(s). - Ilya Gutkovskiy, Apr 18 2021
T(n,k) = Sum_{j=1..n} (n/gcd(n,j))^k*phi(gcd(n,j))/phi(n/gcd(n,j)). - Richard L. Ollerton, May 10 2021
T(n,k) = Sum_{1 <= j_1, j_2, ..., j_k <= n} gcd(j_1, j_2, ..., j_k)^2 = Sum_{d divides n} d * J_k(n/d), where J_k(n) denotes the k-th Jordan totient function. - Peter Bala, Jan 29 2024

A226561 a(n) = Sum_{d|n} d^n * phi(d), where phi(n) is the Euler totient function A000010(n).

Original entry on oeis.org

1, 5, 55, 529, 12501, 94835, 4941259, 67240193, 2324562301, 40039063525, 2853116706111, 35668789979107, 3634501279107037, 66676110291801575, 3503151245145885315, 147575078498173255681, 13235844190181388226833, 236079349222711695887225, 35611553801885644604231623

Offset: 1

Paul D. Hanna, Jun 10 2013

nonn

Compare formula to the identity: Sum_{d|n} phi(d) = n.

			L.g.f.: L(x) = x + 5*x^2/2 + 55*x^3/3 + 529*x^4/4 + 12501*x^5/5 + 94835*x^6/6 + ...
where
exp(L(x)) = 1 + x + 3*x^2 + 21*x^3 + 155*x^4 + 2691*x^5 + 18924*x^6 + 732230*x^7 + 9223166*x^8 + ... + A226560(n)*x^n + ...

Robert Israel, Table of n, a(n) for n = 1..385

Cf. A226560, A226459, A000010, A321349, A332517.

m:=30; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(  (&+[EulerPhi(k)*(k*x)^k/(1-(k*x)^k): k in [1..2*m]]) )); // G. C. Greubel, Nov 07 2018

f:= n -> add(d^n * numtheory:-phi(d), d = numtheory:-divisors(n)):
map(f, [$1..40]); # Robert Israel, Jun 16 2017

Table[DivisorSum[n, #*EulerPhi[#^n]  &], {n, 1, 30}]  (* or *) With[{nmax = 30}, Rest[CoefficientList[Series[Sum[EulerPhi[k]*(k*x)^k/(1 - (k*x)^k), {k, 1, 2*nmax}], {x, 0, nmax}], x]]]  (* G. C. Greubel, Nov 07 2018 *)

{a(n)=sumdiv(n, d, d^n*eulerphi(d))}
for(n=1,30,print1(a(n),", "))

a(n) = sum(k=1, n, (n/gcd(k, n))^n); \\ Seiichi Manyama, Mar 11 2021

from sympy import totient, divisors
def A226561(n):
    return sum(totient(d)*d**n for d in divisors(n,generator=True)) # Chai Wah Wu, Feb 15 2020

Logarithmic derivative of A226560.
a(n) = Sum_{d|n} d * phi(d^n).
a(n) = Sum_{d|n} phi(d^(n+1)).
a(n) = Sum_{d|n} phi(d^(n+2))/d.
a(n) = Sum_{d|n} d^(n-k+1) * phi(d^k) for k >= 1.
G.f.: Sum_{k>=1} phi(k)*(k*x)^k/(1 - (k*x)^k). - Ilya Gutkovskiy, Nov 06 2018
a(n) = Sum_{k=1..n} (n/gcd(k,n))^n. - Seiichi Manyama, Mar 11 2021
a(n) = Sum_{k=1..n} gcd(n,k)^n*phi(gcd(n,k))/phi(n/gcd(n,k)). - Richard L. Ollerton, May 10 2021

A342423 a(n) = Sum_{k=1..n} gcd(k,n)^gcd(k,n).

Original entry on oeis.org

1, 5, 29, 262, 3129, 46693, 823549, 16777484, 387420549, 10000003145, 285311670621, 8916100495490, 302875106592265, 11112006826381589, 437893890380865741, 18446744073726329368, 827240261886336764193, 39346408075296925089309

Offset: 1

Seiichi Manyama, Mar 11 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..386

Cf. A000010, A332517, A342420, A342424.

a[n_] := Sum[GCD[k, n]^GCD[k, n], {k, 1, n}]; Array[a, 18] (* Amiram Eldar, Mar 11 2021 *)

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

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

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

A342433 a(n) = Sum_{k=1..n} gcd(k,n)^(n-1).

Original entry on oeis.org

1, 3, 11, 74, 629, 8085, 117655, 2113796, 43059849, 1001955177, 25937424611, 743379914746, 23298085122493, 793811662313709, 29192938251553759, 1152956691126550536, 48661191875666868497, 2185928270773974154773

Offset: 1

Seiichi Manyama, Mar 12 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..387

Cf. A000010, A000272, A321349, A332517, A332621, A342432, A343510.

a[n_] := Sum[GCD[k, n]^(n - 1), {k, 1, n}]; Array[a, 20] (* Amiram Eldar, Mar 12 2021 *)

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

a(n) = sumdiv(n, d, eulerphi(n/d)*d^(n-1));

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

a(n) = Sum_{d|n} phi(n/d) * d^(n-1).
a(n) = Sum_{d|n} mu(n/d) * d * sigma_(n-2)(d).
a(n) ~ n^(n-1). - Vaclav Kotesovec, May 23 2021

A343499 a(n) = Sum_{k=1..n} gcd(k, n)^5.

Original entry on oeis.org

1, 33, 245, 1058, 3129, 8085, 16813, 33860, 59541, 103257, 161061, 259210, 371305, 554829, 766605, 1083528, 1419873, 1964853, 2476117, 3310482, 4119185, 5315013, 6436365, 8295700, 9778145, 12253065, 14468481, 17788154, 20511177, 25297965, 28629181, 34672912, 39459945, 46855809

Offset: 1

Seiichi Manyama, Apr 17 2021

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

Column 5 of A343510.

Cf. A000010, A001159 (sigma_4(n)), A018804, A054612, A059378, A069097, A332517, A342423, A342433, A343497, A343498, A344524.

A343499:= func< n | (&+[d^5*EulerPhi(Floor(n/d)): d in Divisors(n)]) >;
[A343499(n): n in [1..50]]; // G. C. Greubel, Jun 24 2024

a[n_] := Sum[GCD[k, n]^5, {k, 1, n}]; Array[a, 50] (* Amiram Eldar, Apr 18 2021 *)
f[p_, e_] := p^(e-1)*(p^(4*e+5) - p^(4*e) - p + 1)/(p^4-1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 50] (* Amiram Eldar, Nov 22 2022 *)

PARI
```
a(n) = sum(k=1, n, gcd(k, n)^5);
```

a(n) = sumdiv(n, d, eulerphi(n/d)*d^5);

a(n) = sumdiv(n, d, moebius(n/d)*d*sigma(d, 4));

my(N=40, x='x+O('x^N)); Vec(sum(k=1, N, eulerphi(k)*x^k*(1+26*x^k+66*x^(2*k)+26*x^(3*k)+x^(4*k))/(1-x^k)^6))

def A343499(n): return sum(k^5*euler_phi(n/k) for k in (1..n) if (k).divides(n))
[A343499(n) for n in range(1,51)] # G. C. Greubel, Jun 24 2024

a(n) = Sum_{d|n} phi(n/d) * d^5.
a(n) = Sum_{d|n} mu(n/d) * d * sigma_4(d).
G.f.: Sum_{k >= 1} phi(k) * x^k * (1 + 26*x^k + 66*x^(2*k) + 26*x^(3*k) + x^(4*k))/(1 - x^k)^6.
Dirichlet g.f.: zeta(s-1) * zeta(s-5) / zeta(s). - Ilya Gutkovskiy, Apr 18 2021
Sum_{k=1..n} a(k) ~ 315*zeta(5)*n^6 / (2*Pi^6). - Vaclav Kotesovec, May 20 2021
Multiplicative with a(p^e) = p^(e-1)*(p^(4*e+5) - p^(4*e) - p + 1)/(p^4-1). - Amiram Eldar, Nov 22 2022
a(n) = Sum_{1 <= i_1, ..., i_5 <= n} gcd(i_1, ..., i_5, n) = Sum_{d divides n} d * J_5(n/d), where the Jordan totient function J_5(n) = A059378(n). - Peter Bala, Jan 29 2024

A344223 a(n) = Sum_{k=1..n} tau(gcd(k,n)^n), where tau(n) is the number of divisors of n.

Original entry on oeis.org

1, 4, 6, 16, 10, 72, 14, 64, 45, 180, 22, 600, 26, 336, 360, 256, 34, 1620, 38, 1600, 672, 792, 46, 4752, 175, 1092, 378, 3080, 58, 36960, 62, 1024, 1584, 1836, 1680, 17136, 74, 2280, 2184, 12960, 82, 97020, 86, 7480, 9450, 3312, 94, 37536, 441, 16900, 3672, 10400, 106, 40824, 3960, 25200

Offset: 1

Seiichi Manyama, May 12 2021

nonn

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

Cf. A000203, A332517, A344221, A344222, A344224, A344225, A344226.

Table[Sum[DivisorSigma[0,GCD[k,n]^n],{k,n}],{n,100}] (* Giorgos Kalogeropoulos, May 13 2021 *)

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

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

PARI
```
a(n) = n*sumdiv(n, d, n^omega(d)/d);
```

a(n) = n * A344226(n).
a(n) = Sum_{d|n} phi(n/d) * tau(d^n).
a(n) = n * Sum_{d|n} n^omega(d) / d.
If p is prime, a(p) = 2*p.

A343517 a(n) = Sum_{1 <= x_1 <= x_2 <= ... <= x_n <= n} gcd(x_1, x_2, ... , x_n, n).

Original entry on oeis.org

1, 4, 12, 42, 130, 506, 1722, 6622, 24426, 93427, 352726, 1359388, 5200312, 20097156, 77567064, 300787366, 1166803126, 4539197723, 17672631918, 68933307843, 269129530770, 1052113994340, 4116715363822, 16124224571368, 63205303313900, 247961973949536

Offset: 1

Seiichi Manyama, Apr 17 2021

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 1..1650

Main diagonal of A343516.

Cf. A000010, A156834, A332508, A332517.

a[n_] := DivisorSum[n, EulerPhi[n/#] * Binomial[n + # - 1, n] &]; Array[a, 25] (* Amiram Eldar, Apr 18 2021 *)

a(n) = sumdiv(n, d, eulerphi(n/d)*binomial(d+n-1, n));

a(n) = Sum_{d|n} phi(n/d) * binomial(d+n-1, n).
a(n) = [x^n] Sum_{k >= 1} phi(k) * x^k/(1 - x^k)^(n+1).
a(n) ~ 2^(2*n - 1) / sqrt(Pi*n). - Vaclav Kotesovec, May 23 2021

A342432 a(n) = Sum_{k=1..n} gcd(k,n)^(n-2).

Original entry on oeis.org

1, 2, 5, 22, 129, 1411, 16813, 266372, 4787349, 100391653, 2357947701, 61980047702, 1792160394049, 56707753687079, 1946197516142925, 72061992621375496, 2862423051509815809, 121441389759089405193, 5480386857784802185957

Offset: 1

Seiichi Manyama, Mar 12 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..388

Cf. A000010, A332517, A342412, A342433, A343510.

a[n_] := Sum[GCD[k, n]^(n - 2), {k, 1, n}]; Array[a, 20] (* Amiram Eldar, Mar 12 2021 *)

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

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

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

a(n) = Sum_{d|n} phi(n/d) * d^(n-2).
a(n) = Sum_{d|n} mu(n/d) * d * sigma_(n-3)(d).
a(n) ~ n^(n-2). - Vaclav Kotesovec, May 23 2021

cp's OEIS Frontend

A343497 a(n) = Sum_{k=1..n} gcd(k, n)^3.

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A343498 a(n) = Sum_{k=1..n} gcd(k, n)^4.

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A343510 Square array T(n,k), n >= 1, k >= 1, read by antidiagonals, where T(n,k) = Sum_{j=1..n} gcd(j, n)^k.

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A226561 a(n) = Sum_{d|n} d^n * phi(d), where phi(n) is the Euler totient function A000010(n).

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A342423 a(n) = Sum_{k=1..n} gcd(k,n)^gcd(k,n).

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A342433 a(n) = Sum_{k=1..n} gcd(k,n)^(n-1).

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