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).

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

1,3

Links

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

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).

Example

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, ...

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 *)

Prog

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

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.

Sequence in context: A370807 A298890 A016473 * A366977 A029637 A097207

Adjacent sequences: A343513 A343514 A343515 * A343517 A343518 A343519

Keyword

nonn,tabl

Author

Seiichi Manyama, Apr 17 2021

Status

approved