A357431 -id:A357431 - cp's OEIS frontend

A357417 Row sums of the triangular array A357431.

1, 5, 12, 27, 43, 76, 109, 168, 218, 301, 383, 499, 591, 779, 904, 1153, 1322, 1555, 1817, 2143, 2379, 2790, 3164, 3627, 3957, 4546, 5034, 5599, 6062, 6937, 7456, 8369, 8973, 9896, 10678, 11663, 12430, 13732, 14618, 15920, 16996, 18471, 19570, 20934, 22189, 24080

Offset: 1

Tamas Sandor Nagy, Sep 27 2022

nonn

The rows of the triangular array A357431 are chains of numbers that end with the positive terms of A007952.

It appears that lim_{n->oo} a(n)/A002411(n) will converge to a number close to 0.464401.. . - Thomas Scheuerle, Sep 27 2022

			For n = 6, the numbers of the chain that are divisible by 6, 5, 4, 3, 2, and 1 are 6, 10, 12, 15, 16, and 17, these forming row 6 of A357431. The sum of this row is a(6) = 76.

Cf. A357431, A002411.

function a = A357417( max_n )
    for n = 1:max_n
        k = [n:-1:1];
        for m = 2:length(k)
            k(m) = k(m)*(floor(k(m-1)/k(m))+1);
        end
        a(n) = sum(k);
    end
end % Thomas Scheuerle, Sep 27 2022

a[n_] := Module[{k = n, s = n, r}, Do[k++; k += If[(r = Mod[k, i]) == 0, 0, i - Mod[k, i]]; s += k, {i, n - 1, 1, -1}]; s]; Array[a, 50] (* Amiram Eldar, Sep 27 2022 *)

a(n) = my(t=0); sum(k=0,n-1, t++; t+=(-t)%(n-k)); \\ Kevin Ryde, Sep 27 2022

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

Original entry on oeis.org

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

Tamas Sandor Nagy, Oct 01 2022

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.

			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

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

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

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 *)

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

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).

cp's OEIS Frontend

A357417 Row sums of the triangular array A357431.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

MATLAB

Mathematica

PARI