A321368 Triangular array read by rows: T(n, k) is the number of (i, j), ordered pairs of integers, satisfying 1 <= i < j <= n and floor(n * (1/i - 1/j)) = k; n >= 1, 0 <= k < n.

0, 0, 1, 1, 1, 1, 2, 1, 2, 1, 4, 2, 1, 2, 1, 5, 4, 1, 1, 3, 1, 8, 4, 3, 1, 1, 3, 1, 10, 6, 4, 1, 1, 1, 4, 1, 14, 7, 3, 4, 1, 0, 2, 4, 1, 17, 9, 4, 5, 1, 1, 1, 1, 5, 1, 22, 10, 6, 3, 4, 1, 0, 1, 2, 5, 1, 24, 13, 8, 3, 6, 1, 1, 0, 1, 2, 6, 1, 30, 14, 8, 6, 3, 5

Offset

1,7

Links

Table of n, a(n) for n=1..84.

Luc Rousseau, Colored array plot of T(n,k) for n = 1..300, k = 0..n-1.

Formula

Sum_{k=0..n-1} T(n, k) * k = A321047(n).

Example

Array begins:
   0
   0  1
   1  1  1
   2  1  2  1
   4  2  1  2  1
   5  4  1  1  3  1
   8  4  3  1  1  3  1
  10  6  4  1  1  1  4  1
  14  7  3  4  1  0  2  4  1
  17  9  4  5  1  1  1  1  5  1
  22 10  6  3  4  1  0  1  2  5  1
  24 13  8  3  6  1  1  0  1  2  6  1
  30 14  8  6  3  5  1  0  1  1  2  6  1
  34 18  9  7  2  7  1  1  0  1  1  2  7  1
  40 19 12  8  2  4  6  1  0  0  1  1  3  7  1
  46 21 14  6  7  2  8  1  1  0  1  0  2  2  8  1
  53 24 14  8  8  2  5  6  1  0  0  1  1  1  3  8  1
  58 29 13 11 10  2  3  9  1  1  0  0  1  1  1  3  9  1
  66 30 17 13  7  6  2  5  7  1  0  0  1  0  1  2  3  9  1
  72 34 20 14  5 10  2  3 10  1  1  0  0  1  0  1  2  3 10  1
  80 38 21 13  9 11  2  3  5  8  1  0  0  0  1  1  1  1  4 10  1
  89 42 21 15 11  8  7  2  3 11  1  1  0  0  1  0  1  1  2  3 11  1
  98 44 25 17 12  6 11  1  3  6  8  1  0  0  0  1  0  1  1  2  4 11  1

Mathematica

T[A_, k_] := If[KeyExistsQ[A, k], A[k], 0]
G[n_] := Module[{A = <||>, k, j, i}, For[j = 2, j <= n, j++, For[i = 1, i < j, i++, k = Floor[n*(1/i - 1/j)]; A[k] = T[A, k] + 1]]; A]
For[n = 1, n <= 23, n++, Print[Block[{A = G[n]}, T[A, #] & /@ Range[0, n - 1]]]]

Prog

(Java)
package oeis;
import java.util.*;
public class A {
  private static Map<Integer, Integer> g(int n) {
    Map<Integer, Integer> map = new HashMap<>();
    for (int j = 2; j <= n; j ++) {
      for (int i = 1; i < j; i ++) {
        int k = (n * (j - i)) / (i * j);
        Integer v = map.get(k);
        map.put(k, (v == null) ? 1 : v + 1);
      }
    }
    return map;
  }
  public static void main(String[] args) {
    for (int n = 1; n <= 23; n ++) {
      Map<Integer, Integer> map = g(n);
      for (int k = 0; k < n; k ++) {
        Integer v = map.get(k);
        v = (v == null ? 0 : v);
        System.out.print(String.format("%d, ", v));
      }
      //System.out.println();
    }
  }
}
(PARI) T(n, k) = sum(i=1, n-1, sum(j=i+1, n, floor(n * (1/i - 1/j)) == k));
tabl(nn) = for (i=1, nn, for (j=0, i-1, print1(T(i, j), ", ")); print; ) \\ Michel Marcus, Nov 08 2018

Crossrefs

Cf. A321047.

Sequence in context: A061797 A068341 A360738 * A338508 A100380 A205403

Adjacent sequences: A321365 A321366 A321367 * A321369 A321370 A321371

Keyword

nonn,tabl

Author

Luc Rousseau, Nov 07 2018

Status

approved