cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

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

Original entry on oeis.org

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

Views

Author

Seiichi Manyama, Apr 17 2021

Keywords

Links

Crossrefs

Column 3 of A343510.
Cf. A000010, A001157 (sigma_2(n)), A018804, A054610, A069097, A309323, A332517, A342423, A342433, A343498, A343499, A343513.

Programs

  • Magma
    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
  • Maple
    with(numtheory):
seq(add(phi(n/d) * d^3, d in divisors(n)), n = 1..50); # Peter Bala, Jan 20 2024
  • Mathematica
    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);
  • PARI
    a(n) = sumdiv(n, d, eulerphi(n/d)*d^3);
  • PARI
    a(n) = sumdiv(n, d, moebius(n/d)*d*sigma(d, 2));
  • PARI
    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))
  • SageMath
    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

Formula

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

A343516 Square array T(n,k), n >= 1, k >= 1, read by antidiagonals, where T(n,k) = Sum_{1 <= x_1 <= x_2 <= ... <= x_k <= n} gcd(x_1, x_2, ... , x_k, n).

Original entry on oeis.org

1, 1, 3, 1, 4, 5, 1, 5, 8, 8, 1, 6, 12, 15, 9, 1, 7, 17, 26, 19, 15, 1, 8, 23, 42, 39, 35, 13, 1, 9, 30, 64, 74, 76, 34, 20, 1, 10, 38, 93, 130, 153, 90, 56, 21, 1, 11, 47, 130, 214, 287, 216, 152, 63, 27, 1, 12, 57, 176, 334, 506, 468, 379, 191, 86, 21
Offset: 1

Views

Author

Seiichi Manyama, Apr 17 2021

Keywords

Examples

			T(4,2) = gcd(1,1,4) + gcd(1,2,4) + gcd(2,2,4) + gcd(1,3,4) + gcd(2,3,4) + gcd(3,3,4) + gcd(1,4,4) + gcd(2,4,4) + gcd(3,4,4) + gcd(4,4,4) = 1 + 1 + 2 + 1 + 1 + 1 + 1 + 2 + 1 + 4 = 15.
Square array begins:
   1,  1,  1,   1,   1,   1,    1, ...
   3,  4,  5,   6,   7,   8,    9, ...
   5,  8, 12,  17,  23,  30,   38, ...
   8, 15, 26,  42,  64,  93,  130, ...
   9, 19, 39,  74, 130, 214,  334, ...
  15, 35, 76, 153, 287, 506,  846, ...
  13, 34, 90, 216, 468, 930, 1722, ...

Links

Crossrefs

Columns k=1..7 give A018804, A309322, A309323, A343518, A343519, A343520, A343521.
Main diagonal gives A343517.
T(n,n-1) gives A343553.
Cf. A343510.

Programs

  • Mathematica
    T[n_, k_] := DivisorSum[n, EulerPhi[n/#] * Binomial[k + # - 1, k] &]; Table[T[k, n - k + 1], {n, 1, 11}, {k, 1, n}] // Flatten (* Amiram Eldar, Apr 18 2021 *)
  • PARI
    T(n, k) = sumdiv(n, d, eulerphi(n/d)*binomial(d+k-1, k));

Formula

G.f. of column k: Sum_{j>=1} phi(j) * x^j/(1 - x^j)^(k+1).
T(n,k) = Sum_{d|n} phi(n/d) * binomial(d+k-1, k).

A309322 Expansion of Sum_{k>=1} phi(k) * x^k/(1 - x^k)^3, where phi = Euler totient function (A000010).

Original entry on oeis.org

1, 4, 8, 15, 19, 35, 34, 56, 63, 86, 76, 141, 103, 157, 182, 212, 169, 294, 208, 355, 335, 359, 298, 556, 405, 490, 522, 657, 463, 865, 526, 816, 773, 812, 856, 1239, 739, 1003, 1058, 1424, 901, 1610, 988, 1525, 1617, 1445, 1174, 2188, 1435, 1960, 1760, 2091, 1483, 2529, 1994
Offset: 1

Views

Author

Ilya Gutkovskiy, Jul 23 2019

Keywords

Comments

Dirichlet convolution of Euler totient function with triangular numbers.

Links

Crossrefs

Cf. A000010, A000217, A018804, A069097, A272718, A309323.

Programs

  • Mathematica
    nmax = 55; CoefficientList[Series[Sum[EulerPhi[k] x^k/(1 - x^k)^3, {k, 1, nmax}], {x, 0, nmax}], x] // Rest
Table[Sum[EulerPhi[n/d] d (d + 1)/2, {d, Divisors[n]}], {n, 1, 55}]
Table[Sum[Sum[GCD[j, k, n], {j, 1, k}], {k, 1, n}], {n, 1, 55}]

Formula

a(n) = Sum_{d|n} phi(n/d) * d * (d + 1)/2.
a(n) = Sum_{k=1..n} Sum_{j=1..k} gcd(j,k,n).
a(n) = Sum_{k=1..n} gcd(n,k)*(gcd(n,k)+1)/2. - Richard L. Ollerton, May 07 2021
Sum_{k=1..n} a(k) ~ Pi^2 * n^3 / (36*zeta(3)). - Vaclav Kotesovec, May 23 2021
a(n) = (A018804(n) + A069097(n))/2. - Ridouane Oudra, May 22 2025

A344992 a(n) = Sum_{1 <= i <= j <= k <= m <= n} gcd(i,j,k,m).

Original entry on oeis.org

1, 6, 18, 44, 83, 159, 249, 401, 592, 867, 1163, 1655, 2122, 2796, 3594, 4594, 5579, 7046, 8394, 10328, 12339, 14699, 17021, 20441, 23526, 27317, 31379, 36323, 40846, 47300, 52786, 59954, 67191, 75380, 83720, 94662, 103837, 115137, 126851, 141059, 153440
Offset: 1

Views

Author

Vaclav Kotesovec, Jun 05 2021

Keywords

Comments

In general, if g.f.: 1/(1-x) * Sum_{j>=1} phi(j) * x^j/(1 - x^j)^k, where k > 2 and phi is the Euler totient function (A000010), then a(n) ~ zeta(k-1) * n^k / (k! * zeta(k)).

Links

Crossrefs

Column k=4 of A345229.
Partial sums of A309323.
Cf. A000010, A059358, A272718, A309322, A344521.

Programs

  • Mathematica
    Table[Sum[Sum[Sum[Sum[GCD[i, j, k, m], {i, 1, j}], {j, 1, k}], {k, 1, m}], {m, 1, n}], {n, 1, 100}]
nmax = 100; Rest[CoefficientList[Series[1/(1-x) * Sum[EulerPhi[k]*x^k/(1 - x^k)^4, {k, 1, nmax}], {x, 0, nmax}], x]]
Accumulate[Table[Sum[EulerPhi[n/d] * d*(d+1)*(d+2)/6, {d, Divisors[n]}], {n, 1, 100}]] (* faster *)
  • PARI
    a(n) = sum(i=1, n, sum(j=i, n, sum(k=j, n, sum(m=k, n, gcd([i, j, k, m]))))); \\ Michel Marcus, Jun 06 2021

Formula

G.f.: 1/(1-x) * Sum_{k>=1} phi(k) * x^k/(1 - x^k)^4, where phi is the Euler totient function (A000010).
a(n) = Sum_{k=1..n} Sum_{d|k} phi(k/d) * d*(d+1)*(d+2)/6.
a(n) ~ 15 * zeta(3) * n^4 / (4*Pi^4).
a(n) = Sum_{k=1..n} phi(k) * binomial(floor(n/k)+3,4). - Seiichi Manyama, Sep 13 2024
Showing 1-4 of 4 results.

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