A344992 -id:A344992 - cp's OEIS frontend

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).

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

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

Seiichi Manyama, May 22 2021

nonn

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

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

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 *)

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

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

cp's OEIS Frontend

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).

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

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