A346928 Irregular triangle read by rows; the n-th row contains, in ascending order, the distinct integers of the form n! / m! (with 1 <= m <= n) that do not appear in former rows.

1, 2, 3, 6, 4, 12, 24, 5, 20, 60, 120, 30, 360, 720, 7, 42, 210, 840, 2520, 5040, 8, 56, 336, 1680, 6720, 20160, 40320, 9, 72, 504, 3024, 15120, 60480, 181440, 362880, 10, 90, 30240, 151200, 604800, 1814400, 3628800, 11, 110, 990, 7920, 55440, 332640, 1663200, 6652800, 19958400, 39916800

Offset: 1

Author

Rémy Sigrist, Oct 16 2021

Keywords

• nonn
• tabf

Comments

This sequence has similarities with A209260; here we consider quotients of factorial numbers, there differences of triangular numbers.

The n-th row is included in the n-th row of A068424 and has greatest term n!.

As a flat sequence, we have a permutation of the positive integers (any n > 0 appears among the first n rows, see A348401).

For any prime number p, the p-th row contains p-1 terms.

Examples

			Triangle begins:
    1;
    2;
    3, 6;
    4, 12, 24;
    5, 20, 60, 120;
    30, 360, 720;
    7, 42, 210, 840, 2520, 5040;
    8, 56, 336, 1680, 6720, 20160, 40320;
    9, 72, 504, 3024, 15120, 60480, 181440, 362880;
    10, 90, 30240, 151200, 604800, 1814400, 3628800;
    11, 110, 990, 7920, 55440, 332640, 1663200, 6652800, 19958400, 39916800;
    ...
		

Links

• Index entries for sequences related to factorial numbers
• Index entries for sequences that are permutations of the natural numbers

Crossrefs

Cf. A000142, A068424, A209260, A348401.

Programs

PARI

s=[]; for (n=1, 11, p=1; forstep (m=n, 1, -1, if (!setsearch(s, p*=m), s=setunion(s, [p]); print1 (p", "))))

Python

from math import factorial
def auptor(rows):
    alst, aset = [1], {1}
    for n in range(2, rows+1):
        fn = factorial(n)
        for m in range(n-1, 0, -1):
            fm = factorial(m)
            q, r = divmod(fn, factorial(m))
            if r == 0 and q not in aset:
                alst.append(q); aset.add(q)
    return alst
print(auptor(11)) # Michael S. Branicky, Oct 17 2021