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

1,2

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.

Links

Table of n, a(n) for n=1..52.

Index entries for sequences related to factorial numbers

Index entries for sequences that are permutations of the natural numbers

Example

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;
    ...

Prog

(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

Crossrefs

Cf. A000142, A068424, A209260, A348401.

Sequence in context: A130879 A119741 A268216 * A245712 A285331 A237125

Adjacent sequences: A346925 A346926 A346927 * A346929 A346930 A346931

Keyword

nonn,tabf

Author

Rémy Sigrist, Oct 16 2021

Status

approved