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 1-5 of 5 results.

A344522 a(n) = Sum_{1 <= i, j, k <= n} gcd(i,j,k).

Original entry on oeis.org

1, 9, 30, 76, 141, 267, 400, 624, 885, 1249, 1590, 2208, 2689, 3411, 4248, 5248, 6081, 7485, 8530, 10248, 11889, 13687, 15228, 17988, 20053, 22569, 25242, 28588, 31053, 35463, 38284, 42540, 46581, 50893, 55362, 61824, 65857, 71247, 76884, 84388, 89349, 97881, 103342
Offset: 1

Views

Author

Seiichi Manyama, May 22 2021

Keywords

Links

Crossrefs

Column k=3 of A344479.
Cf. A018806, A071778, A343497, A344132, A344521, A344523, A344524, A344525, A344526.

Programs

  • Mathematica
    a[n_] := Sum[EulerPhi[k] * Quotient[n, k]^3, {k, 1, n}]; Array[a, 50] (* Amiram Eldar, May 22 2021 *)
  • PARI
    a(n) = sum(i=1, n, sum(j=1, n, sum(k=1, n, gcd([i, j, k]))));
  • PARI
    a(n) = sum(k=1, n, eulerphi(k)*(n\k)^3);
  • PARI
    my(N=66, x='x+O('x^N)); Vec(sum(k=1, N, eulerphi(k)*x^k*(1+4*x^k+x^(2*k))/(1-x^k)^3)/(1-x))

Formula

a(n) = Sum_{k=1..n} phi(k) * floor(n/k)^3.
G.f.: (1/(1 - x)) * Sum_{k >= 1} phi(k) * x^k * (1 + 4*x^k + x^(2*k))/(1 - x^k)^3.
a(n) ~ Pi^2 * n^3 / (6*zeta(3)). - Vaclav Kotesovec, May 23 2021

A344479 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).

Original entry on oeis.org

1, 1, 3, 1, 5, 6, 1, 9, 12, 10, 1, 17, 30, 24, 15, 1, 33, 84, 76, 37, 21, 1, 65, 246, 276, 141, 61, 28, 1, 129, 732, 1060, 649, 267, 80, 36, 1, 257, 2190, 4164, 3165, 1417, 400, 112, 45, 1, 513, 6564, 16516, 15697, 8091, 2528, 624, 145, 55, 1, 1025, 19686, 65796, 78261, 47521, 17128, 4432, 885, 189, 66
Offset: 1

Views

Author

Seiichi Manyama, May 22 2021

Keywords

Examples

			G.f. of column 3: (1/(1 - x)) * Sum_{i>=1} phi(i) * (x^i + 4*x^(2*i) + x^(3*i))/(1 - x^i)^3.
Square array begins:
   1,  1,   1,    1,    1,     1, ...
   3,  5,   9,   17,   33,    65, ...
   6, 12,  30,   84,  246,   732, ...
  10, 24,  76,  276, 1060,  4164, ...
  15, 37, 141,  649, 3165, 15697, ...
  21, 61, 267, 1417, 8091, 47521, ...

Links

Crossrefs

Columns k=1..5 give A000217, A018806, A344522, A344523, A344524.
T(n,n) gives A344525.
Cf. A343510, A343516, A344527.

Programs

  • Mathematica
    T[n_, k_] := Sum[EulerPhi[j] * Quotient[n, j]^k, {j, 1, n}]; Table[T[k, n - k + 1], {n, 1, 11}, {k, 1, n}] // Flatten (* Amiram Eldar, May 22 2021 *)
  • PARI
    T(n, k) = sum(j=1, n, eulerphi(j)*(n\j)^k);

Formula

G.f. of column k: (1/(1 - x)) * Sum_{i>=1} phi(i) * ( Sum_{j=1..k} A008292(k, j) * x^(i*j) )/(1 - x^i)^k.
T(n,k) = Sum_{j=1..n} phi(j) * floor(n/j)^k.

A344523 a(n) = Sum_{1 <= i, j, k, l <= n} gcd(i,j,k,l).

Original entry on oeis.org

1, 17, 84, 276, 649, 1417, 2528, 4432, 7033, 10905, 15556, 22836, 30673, 41729, 54944, 71968, 89969, 115457, 140820, 175444, 212537, 257113, 302720, 366160, 426505, 500873, 580676, 677108, 769761, 895377, 1008928, 1153120, 1300417, 1469073, 1640020, 1860340, 2054921
Offset: 1

Views

Author

Seiichi Manyama, May 22 2021

Keywords

Links

Crossrefs

Column k=4 of A344479.
Cf. A082540, A343498, A344138, A344522, A344524, A344525.

Programs

  • Mathematica
    a[n_] := Sum[EulerPhi[k] * Quotient[n, k]^4, {k, 1, n}]; Array[a, 50] (* Amiram Eldar, May 22 2021 *)
  • PARI
    a(n) = sum(i=1, n, sum(j=1, n, sum(k=1, n, sum(l=1, n, gcd([i, j, k, l])))));
  • PARI
    a(n) = sum(k=1, n, eulerphi(k)*(n\k)^4);
  • PARI
    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)^4)/(1-x))

Formula

a(n) = Sum_{k=1..n} phi(k) * floor(n/k)^4.
G.f.: (1/(1 - x)) * Sum_{k >= 1} phi(k) * x^k * (1 + 11*x^k + 11*x^(2*k) + x^(3*k))/(1 - x^k)^4.
a(n) ~ 90 * zeta(3) * n^4 / Pi^4. - Vaclav Kotesovec, May 23 2021

A344524 a(n) = Sum_{1 <= i, j, k, l, m <= n} gcd(i,j,k,l,m).

Original entry on oeis.org

1, 33, 246, 1060, 3165, 8091, 17128, 33936, 60645, 103825, 164886, 259368, 381841, 557595, 784200, 1091056, 1462353, 1968261, 2554810, 3327120, 4230561, 5361463, 6644196, 8302020, 10113445, 12352041, 14873418, 17924356, 21225165, 25341375, 29670556, 34920348, 40625541, 47297365
Offset: 1

Views

Author

Seiichi Manyama, May 22 2021

Keywords

Comments

In general, for m > 2, Sum_{k=1..n} phi(k) * floor(n/k)^m ~ zeta(m-1) * n^m / zeta(m). - Vaclav Kotesovec, May 23 2021

Links

Crossrefs

Column k=5 of A344479.
Cf. A082544, A343499, A344139, A344522, A344523, A344525.

Programs

  • Mathematica
    a[n_] := Sum[EulerPhi[k] * Quotient[n, k]^5, {k, 1, n}]; Array[a, 50] (* Amiram Eldar, May 22 2021 *)
  • PARI
    a(n) = sum(i=1, n, sum(j=1, n, sum(k=1, n, sum(l=1, n, sum(m=1, n, gcd([i, j, k, l, m]))))));
  • PARI
    a(n) = sum(k=1, n, eulerphi(k)*(n\k)^5);
  • PARI
    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)^5)/(1-x))

Formula

a(n) = Sum_{k=1..n} phi(k) * floor(n/k)^5.
G.f.: (1/(1 - x)) * 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)^5.
a(n) ~ Pi^4 * n^5 / (90*zeta(5)). - Vaclav Kotesovec, May 23 2021

A345230 a(n) = Sum_{1 <= x_1 <= x_2 <= ... <= x_n <= n} gcd(x_1, x_2, ..., x_n).

Original entry on oeis.org

0, 1, 4, 13, 44, 140, 512, 1782, 6652, 24682, 93599, 354341, 1359470, 5210328, 20098886, 77621774, 300797854, 1167164438, 4539201401, 17674941735, 68933414989, 269143872226, 1052114789548, 4116808923486, 16124224585644, 63205911146740, 247961982954952
Offset: 0

Views

Author

Seiichi Manyama, Jun 11 2021

Keywords

Links

Crossrefs

Main diagonal of A345229.
Cf. A343517, A343553, A344525.

Programs

  • Maple
    a:= n-> coeff(series((1/(1-x))* add(numtheory[phi](k)
         *x^k/(1-x^k)^n, k=1..n), x, n+1), x, n):
seq(a(n), n=0..26);  # Alois P. Heinz, Jun 11 2021
  • Mathematica
    a[n_] := Sum[DivisorSum[k, EulerPhi[k/#] * Binomial[n + # - 2, n - 1] &], {k, 1, n}]; Array[a, 30, 0] (* Amiram Eldar, Jun 11 2021 *)
  • PARI
    a(n) = sum(k=1, n, sumdiv(k, d, eulerphi(k/d)*binomial(d+n-2, n-1)));
  • PARI
    a(n) = sum(k=1, n, eulerphi(k)*binomial(n\k+n-1, n)); \\ Seiichi Manyama, Sep 13 2024

Formula

a(n) = Sum_{k=1..n} Sum_{d|k} phi(k/d) * binomial(d+n-2, n-1).
a(n) = [x^n] (1/(1 - x)) * Sum_{k >= 1} phi(k) * x^k/(1 - x^k)^n.
a(n) ~ 2^(2*n-1) / sqrt(Pi*n). - Vaclav Kotesovec, Jun 11 2021
a(n) = Sum_{k=1..n} phi(k) * binomial(floor(n/k)+n-1,n). - Seiichi Manyama, Sep 13 2024
Showing 1-5 of 5 results.

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