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.

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).

A309323 Expansion of Sum_{k>=1} phi(k) * x^k/(1 - x^k)^4, where phi = Euler totient function (A000010).

Original entry on oeis.org

1, 5, 12, 26, 39, 76, 90, 152, 191, 275, 296, 492, 467, 674, 798, 1000, 985, 1467, 1348, 1934, 2011, 2360, 2322, 3420, 3085, 3791, 4062, 4944, 4523, 6454, 5486, 7168, 7237, 8189, 8340, 10942, 9175, 11300, 11714, 14208, 12381, 16759, 14232, 18036, 18549, 19706, 18470
Offset: 1

Views

Author

Ilya Gutkovskiy, Jul 23 2019

Keywords

Comments

Dirichlet convolution of Euler totient function with tetrahedral numbers.

Links

Crossrefs

Cf. A000010, A000292, A018804, A272718, A309322.

Programs

  • Mathematica
    nmax = 47; CoefficientList[Series[Sum[EulerPhi[k] x^k/(1 - x^k)^4, {k, 1, nmax}], {x, 0, nmax}], x] // Rest
Table[Sum[EulerPhi[n/d] d (d + 1) (d + 2)/6, {d, Divisors[n]}], {n, 1, 47}]
Table[Sum[Sum[Sum[GCD[i, j, k, n], {i, 1, j}], {j, 1, k}], {k, 1, n}], {n, 1, 47}]

Formula

a(n) = Sum_{d|n} phi(n/d) * d * (d + 1) * (d + 2)/6.
a(n) = Sum_{k=1..n} Sum_{j=1..k} Sum_{i=1..j} gcd(i,j,k,n).
Sum_{k=1..n} a(k) ~ 15 * zeta(3) * n^4 / (4*Pi^4). - Vaclav Kotesovec, May 23 2021

A344521 a(n) = Sum_{1 <= i <= j <= k <= n} gcd(i,j,k).

Original entry on oeis.org

1, 5, 13, 28, 47, 82, 116, 172, 235, 321, 397, 538, 641, 798, 980, 1192, 1361, 1655, 1863, 2218, 2553, 2912, 3210, 3766, 4171, 4661, 5183, 5840, 6303, 7168, 7694, 8510, 9283, 10095, 10951, 12190, 12929, 13932, 14990, 16414, 17315, 18925, 19913, 21438, 23055, 24500, 25674, 27862
Offset: 1

Views

Author

Seiichi Manyama, May 22 2021

Keywords

Links

Crossrefs

Column k=3 of A345229.
Partial sums of A309322.
Cf. A272718, A344132, A344522, A344992.

Programs

  • Mathematica
    a[n_] := Sum[Sum[Sum[GCD[i, j, k], {i, 1, j}], {j, 1, k}], {k, 1, n}]; Array[a, 50] (* Amiram Eldar, May 25 2021 *)
nmax = 100; Rest[CoefficientList[Series[1/(1 - x)*Sum[EulerPhi[k]*x^k/(1 - x^k)^3, {k, 1, nmax}], {x, 0, nmax}], x]] (* Vaclav Kotesovec, Jun 05 2021 *)
Accumulate[Table[Sum[EulerPhi[n/d] * d*(d+1)/2, {d, Divisors[n]}], {n, 1, 100}]] (* Vaclav Kotesovec, Jun 05 2021 *)
  • PARI
    a(n) = sum(i=1, n, sum(j=i, n, sum(k=j, n, gcd([i, j, k]))));

Formula

From Vaclav Kotesovec, Jun 05 2021: (Start)
a(n) ~ Pi^2 * n^3 / (36*zeta(3)).
G.f.: 1/(1-x) * Sum_{k>=1} phi(k) * x^k/(1 - x^k)^3, where phi is the Euler totient function (A000010).
a(n) = Sum_{k=1..n} Sum_{d|k} phi(k/d) * d*(d+1)/2. (End)
a(n) = Sum_{k=1..n} phi(k) * binomial(floor(n/k)+2,3). - Seiichi Manyama, Sep 13 2024

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.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.