A344521 -id:A344521 - cp's OEIS frontend

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

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

Seiichi Manyama, May 22 2021

nonn

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

Column k=3 of A344479.

Cf. A018806, A071778, A343497, A344132, A344521, A344523, A344524, A344525, A344526.

a[n_] := Sum[EulerPhi[k] * Quotient[n, k]^3, {k, 1, n}]; Array[a, 50] (* Amiram Eldar, May 22 2021 *)

a(n) = sum(i=1, n, sum(j=1, n, sum(k=1, n, gcd([i, j, k]))));

a(n) = sum(k=1, n, eulerphi(k)*(n\k)^3);

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))

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

A345229 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, 4, 6, 1, 5, 9, 10, 1, 6, 13, 17, 15, 1, 7, 18, 28, 26, 21, 1, 8, 24, 44, 47, 41, 28, 1, 9, 31, 66, 83, 82, 54, 36, 1, 10, 39, 95, 140, 159, 116, 74, 45, 1, 11, 48, 132, 225, 293, 249, 172, 95, 55, 1, 12, 58, 178, 346, 512, 509, 401, 235, 122, 66, 1, 13, 69, 234, 512, 852, 980, 888, 592, 321, 143, 78

Offset: 1

Seiichi Manyama, Jun 11 2021

			G.f. of column 3: (1/(1 - x)) * Sum_{j>=1} phi(j) * x^j/(1 - x^j)^3.
Square array begins:
   1,  1,   1,   1,   1,   1,    1, ...
   3,  4,   5,   6,   7,   8,    9, ...
   6,  9,  13,  18,  24,  31,   39, ...
  10, 17,  28,  44,  66,  95,  132, ...
  15, 26,  47,  83, 140, 225,  346, ...
  21, 41,  82, 159, 293, 512,  852, ...
  28, 54, 116, 249, 509, 980, 1782, ...

Seiichi Manyama, Antidiagonals n = 1..140, flattened

Columns k=1..4 give A000217, A272718, A344521, A344992.
Main diagonal gives A345230.
Cf. A343516, A344479.

T:= (n, k)-> coeff(series((1/(1-x))* add(numtheory[phi](j)
             *x^j/(1-x^j)^k, j=1..n), x, n+1), x, n):
seq(seq(T(n, 1+d-n), n=1..d), d=1..12);  # Alois P. Heinz, Jun 11 2021

T[n_, k_] := Sum[DivisorSum[j, EulerPhi[j/#] * Binomial[k + # - 2, k - 1] &], {j, 1, n}];  Table[T[k, n - k + 1], {n, 1, 12}, {k, 1, n}] // Flatten (* Amiram Eldar, Jun 11 2021 *)

T(n, k) = sum(j=1, n, sumdiv(j, d, eulerphi(j/d)*binomial(d+k-2, k-1)));

T(n, k) = sum(j=1, n, eulerphi(j)*binomial(n\j+k-1, k)); \\ Seiichi Manyama, Sep 13 2024

G.f. of column k: (1/(1 - x)) * Sum_{j>=1} phi(j) * x^j/(1 - x^j)^k.
T(n,k) = Sum_{j=1..n} Sum_{d|j} phi(j/d) * binomial(d+k-2, k-1).
T(n,k) = Sum_{j=1..n} phi(j) * binomial(floor(n/j)+k-1,k). - 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

Vaclav Kotesovec, Jun 05 2021

nonn

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)).

Vaclav Kotesovec, Table of n, a(n) for n = 1..10000

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

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 *)

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

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

cp's OEIS Frontend

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

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

PARI

PARI

Formula

A345229 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).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

PARI

Formula

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula