A389099 Array read by ascending antidiagonals: A(n, k) = k / gcd(n^k, k).

0, 0, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 1, 3, 1, 0, 1, 2, 3, 4, 1, 0, 1, 1, 1, 1, 5, 1, 0, 1, 2, 3, 4, 5, 6, 1, 0, 1, 1, 3, 1, 5, 3, 7, 1, 0, 1, 2, 1, 4, 5, 2, 7, 8, 1, 0, 1, 1, 3, 1, 1, 3, 7, 1, 9, 1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 1, 0, 1, 1, 1, 1, 5, 1, 7, 1, 1, 5, 11, 1

Offset

0,9

Comments

Consider the case n = 2, the general case is analogous: odd(k) = k / gcd(2^k, k) is the odd part of k (A000265), but defined for k >= 0. The even part of k is gcd(2^k, k) (A006519) and also defined for k >= 0. This representation of k = even(k) * odd(k) avoids the restrictions that arise when using the p-adic order.

From Peter Bala, Oct 18 2025: (Start)

The k-th column sequence is a periodic sequence of period rad(k) = A007947(k).

Equivalently, the g.f. of the k-th column sequence is a quasi-polynomial in n with rad(k) constituent polynomials of degree 0.

For example, the 48-th column, beginning [1, 48, 3, 16, 3, 48, 1, 48, 3, 16, 3, 48, 1, 48, 3, 16, 3, 48, ...], has period 6 = rad(48). (End)

Links

Table of n, a(n) for n=0..90.

Peter Bala, A note on the sequence of numerators of a rational function.

Wikipedia, Quasi-polynomial.

Formula

A(n, k) * A389101(n, k) = k for all n >= 0.

From Peter Bala, Oct 18 2025: (Start)

For n > 0, A(n, k) = numerator(k/n^k).

A(n, k) = 1 if and only if n is a multiple of rad(k) = A007947(k).

If k is squarefree (see A005117) then A(n, k) = k/gcd(n, k); if k is cubefree (see A004709) then A(n, k) = k/gcd(n^2, k) and so on. (End)

Example

Array starts:
  [0]  0, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
  [1]  0, 1, 2, 3, 4, 5, 6, 7, 8, 9, ...
  [2]  0, 1, 1, 3, 1, 5, 3, 7, 1, 9, ...
  [3]  0, 1, 2, 1, 4, 5, 2, 7, 8, 1, ...
  [4]  0, 1, 1, 3, 1, 5, 3, 7, 1, 9, ...
  [5]  0, 1, 2, 3, 4, 1, 6, 7, 8, 9, ...
  [6]  0, 1, 1, 1, 1, 5, 1, 7, 1, 1, ...
  [7]  0, 1, 2, 3, 4, 5, 6, 1, 8, 9, ...
  [8]  0, 1, 1, 3, 1, 5, 3, 7, 1, 9, ...
  [9]  0, 1, 2, 1, 4, 5, 2, 7, 8, 1, ...
Seen as a triangle:
  [0] 0;
  [1] 0, 1;
  [2] 0, 1, 1;
  [3] 0, 1, 2, 1;
  [4] 0, 1, 1, 3, 1;
  [5] 0, 1, 2, 3, 4, 1;
  [6] 0, 1, 1, 1, 1, 5, 1;
  [7] 0, 1, 2, 3, 4, 5, 6, 1;
  [8] 0, 1, 1, 3, 1, 5, 3, 7, 1;
  [9] 0, 1, 2, 1, 4, 5, 2, 7, 8, 1;

Maple

A := (n, k) -> k / igcd(n^k, k):
seq(print(seq(A(n, k), k = 0..9)), n = 0..9);

Mathematica

Flatten@ Table[If[k == 0, 0, k/GCD[PowerMod[n, k, k], k]], {n, 0, 12}, {k, 0, n}] (* Michael De Vlieger, Sep 29 2025 *)

Prog

(SageMath)
def A(n: int, k: int) -> int:
    return k // gcd(n ^ k, k)
for n in (0..9): print([A(n, k) for k in (0..9)])

Crossrefs

Rows: A057427, A001477, A000265, A038502, A132739, A065330, A242603.

Columns: A000004, A000012, A000034, A169609, A010685, A171372, A129203, A010689, A166925.

Cf. A279911 (row sums of triangle), A389101, A007947, A005117, A004709.

Sequence in context: A131255 A344566 A198295 * A221857 A133607 A103631

Adjacent sequences: A389096 A389097 A389098 * A389100 A389101 A389102

Keyword

nonn,tabl

Author

Peter Luschny, Sep 29 2025

Status

approved