A343510 Square array T(n,k), n >= 1, k >= 1, read by antidiagonals, where T(n,k) = Sum_{j=1..n} gcd(j, n)^k.

1, 1, 3, 1, 5, 5, 1, 9, 11, 8, 1, 17, 29, 22, 9, 1, 33, 83, 74, 29, 15, 1, 65, 245, 274, 129, 55, 13, 1, 129, 731, 1058, 629, 261, 55, 20, 1, 257, 2189, 4162, 3129, 1411, 349, 92, 21, 1, 513, 6563, 16514, 15629, 8085, 2407, 596, 105, 27, 1, 1025, 19685, 65794, 78129, 47515, 16813, 4388, 789, 145, 21

Offset

1,3

Links

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

Peter Bala, GCD sum theorems. Two Multivariable Cesaro Type Identities.

Eric Weisstein's World of Mathematics, Eulerian Number and Euler's Number Triangle

Formula

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

T(n,k) = Sum_{d|n} phi(n/d) * d^k.

T(n,k) = Sum_{d|n} mu(n/d) * d * sigma_{k-1}(d).

Dirichlet g.f. of column k: zeta(s-1) * zeta(s-k) / zeta(s). - Ilya Gutkovskiy, Apr 18 2021

T(n,k) = Sum_{j=1..n} (n/gcd(n,j))^k*phi(gcd(n,j))/phi(n/gcd(n,j)). - Richard L. Ollerton, May 10 2021

T(n,k) = Sum_{1 <= j_1, j_2, ..., j_k <= n} gcd(j_1, j_2, ..., j_k)^2 = Sum_{d divides n} d * J_k(n/d), where J_k(n) denotes the k-th Jordan totient function. - Peter Bala, Jan 29 2024

Example

G.f. of column 3: Sum_{i>=1} phi(i) * (x^i + 4*x^(2*i) + x^(3*i))/(1 - x^i)^4.
Square array begins:
   1,  1,   1,    1,     1,      1,      1, ...
   3,  5,   9,   17,    33,     65,    129, ...
   5, 11,  29,   83,   245,    731,   2189, ...
   8, 22,  74,  274,  1058,   4162,  16514, ...
   9, 29, 129,  629,  3129,  15629,  78129, ...
  15, 55, 261, 1411,  8085,  47515, 282381, ...
  13, 55, 349, 2407, 16813, 117655, 823549, ...

Mathematica

T[n_, k_] := DivisorSum[n, EulerPhi[n/#] * #^k &]; Table[T[k, n - k + 1], {n, 1, 11}, {k, 1, n}] // Flatten (* Amiram Eldar, Apr 18 2021 *)

Prog

(PARI) T(n, k) = sum(j=1, n, gcd(j, n)^k);
(PARI) T(n, k) = sumdiv(n, d, eulerphi(n/d)*d^k);
(PARI) T(n, k) = sumdiv(n, d, moebius(n/d)*d*sigma(d, k-1));

Crossrefs

Columns k=1..7 give A018804, A069097, A343497, A343498, A343499, A343508, A343509.

T(n-2,n) gives A342432.

T(n-1,n) gives A342433.

T(n,n) gives A332517.

T(n,n+1) gives A321294.

Cf. A008292, A343516.

Sequence in context: A209159 A182397 A376102 * A344725 A209560 A211977

Adjacent sequences: A343507 A343508 A343509 * A343511 A343512 A343513

Keyword

nonn,tabl,easy

Author

Seiichi Manyama, Apr 17 2021

Status

approved