A357498 Triangle read by rows where each term in row n is the next greater multiple of n..1 divided by n..1.

1, 1, 3, 1, 2, 5, 1, 2, 4, 9, 1, 2, 3, 5, 11, 1, 2, 3, 5, 8, 17, 1, 2, 3, 4, 6, 10, 21, 1, 2, 3, 4, 6, 9, 14, 29, 1, 2, 3, 4, 5, 7, 10, 16, 33, 1, 2, 3, 4, 5, 7, 9, 13, 20, 41, 1, 2, 3, 4, 5, 6, 8, 11, 15, 23, 47, 1, 2, 3, 4, 5, 6, 8, 10, 13, 18, 28, 57

Offset: 1

Author

Tamas Sandor Nagy, Oct 01 2022

Keywords

• nonn
• tabl
• easy

Comments

This triangle is related to A357431. Terms there are divisible by n..1 and here that division is performed, leaving the respective multiple of each.

Row n has length n and columns are numbered k = 1..n corresponding to multiples n..1.

Row n begins with n/n = 1. The end-most terms of the rows are A007952.

Examples

			Triangle begins:
  n/k|  1   2   3   4   5   6   7
  --------------------------------
  1  |  1;
  2  |  1,  3;
  3  |  1,  2,  5;
  4  |  1,  2,  4,  9;
  5  |  1,  2,  3,  5, 11;
  6  |  1,  2,  3,  5,  8, 17;
  7  |  1,  2,  3,  4,  6, 10, 21;
  ...
For row n=6, we have:
  A357431 row  6 10 12 15 16 17
  divided by   6  5  4  3  2  1
  results in   1  2  3  5  8 17
		

Links

• Neal Gersh Tolunsky, Table of n, a(n) for n = 1..10011 (141 rows, flattened)

Crossrefs

Cf. A358435 (row sums), A357431, A007952 (right diagonal).

Programs

Mathematica

row[n_] := Module[{k = n, s = Table[0, n], r}, s[[1]] = 1; Do[k++; k += If[(r = Mod[k, i]) == 0, 0, i - Mod[k, i]]; s[[n + 1 - i]] = k/i, {i, n - 1, 1, -1}]; s]; Array[row, 12] // Flatten (* Amiram Eldar, Oct 01 2022 *)

PARI

row(n) = my(v=vector(n)); v[1] = n; for (k=2, n, v[k] = v[k-1] + (n-k+1) - (v[k-1] % (n-k+1));); vector(n, k, v[k]/(n-k+1)); \\ Michel Marcus, Nov 16 2022

Formula

T(n,k) = A357431(n,k) / (n-k+1).

T(n,1) = 1.

T(n,k) = (T(n,k-1)*(n-k+2) + (n-k+1) - (T(n,k-1)*(n-k+2)) mod (n-k+1))/(n-k+1), for k >= 2.

T(n,n) = A007952(n).