A360298 Irregular triangle (an infinite binary tree) read by rows. The tree has root node 1 in row n = 1. For n > 1, each node with value m in row n-1 has a left child with value m / n if n divides m, and a right child with value m * n.

1, 2, 6, 24, 120, 20, 720, 140, 5040, 1120, 630, 40320, 10080, 70, 5670, 4480, 362880, 1008, 100800, 7, 700, 567, 56700, 448, 44800, 36288, 3628800, 11088, 1108800, 77, 7700, 6237, 623700, 4928, 492800, 399168, 39916800, 924, 133056, 92400, 13305600, 924, 92400, 74844, 51975, 7484400, 59136, 5913600, 33264, 4790016, 3326400, 479001600

Offset

1,2

Comments

This sequence is a variant of A360173; here we use divisions and multiplications, there subtractions and additions.

The n-th row has A360299(n) terms, starts with A008336(n+1) and ends with A000142(n).

Links

Rémy Sigrist, Table of n, a(n) for n = 1..11270 (rows for n = 1..26 flattened)

Formula

T(n, 1) = A008336(n+1).

T(n, A360299(n)) = A000142(n).

T(p, k) = p * T(p-1, k) for any prime number p.

Example

The tree begins:
  n     n-th row
  --    --------
   1    1___
            |
   2        2___
                |
   3            6___
                    |
   4               24___
                        |
   5      _____________120_____________
         |                             |
   6    20___                         720___
             |                              |
   7        140___                   _____5040_____
                  |                 |              |
   8            1120__           __630__       __40320__
                      |         |       |     |         |
   9                10080      70     5670  4480     362880

Prog

(PARI) row(n) = { my (r = [1]); for (h = 2, n, r=concat(apply(v -> if (v%h==0, [v/h, v*h], [v*h]), r))); return (r) }
(Python)
from functools import cache
@cache
def row(n):
    if n == 1: return [1]
    out = []
    for r in row(n-1): out += ([r//n] if r%n == 0 else []) + [r*n]
    return out
print([an for r in range(1, 13) for an in row(r)]) # Michael S. Branicky, Feb 02 2023

Crossrefs

Cf. A000142, A008336, A360173, A360299 (row lengths), A360300.

Sequence in context: A065422 A260850 A008336 * A033643 A050211 A248766

Adjacent sequences: A360295 A360296 A360297 * A360299 A360300 A360301

Keyword

nonn,look,tabf,easy

Author

Rémy Sigrist, Feb 02 2023

Status

approved