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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.

%I A346928 #13 Oct 17 2021 15:25:18
%S A346928 1,2,3,6,4,12,24,5,20,60,120,30,360,720,7,42,210,840,2520,5040,8,56,
%T A346928 336,1680,6720,20160,40320,9,72,504,3024,15120,60480,181440,362880,10,
%U A346928 90,30240,151200,604800,1814400,3628800,11,110,990,7920,55440,332640,1663200,6652800,19958400,39916800
%N 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.
%C A346928 This sequence has similarities with A209260; here we consider quotients of factorial numbers, there differences of triangular numbers.
%C A346928 The n-th row is included in the n-th row of A068424 and has greatest term n!.
%C A346928 As a flat sequence, we have a permutation of the positive integers (any n > 0 appears among the first n rows, see A348401).
%C A346928 For any prime number p, the p-th row contains p-1 terms.
%H A346928 <a href="/index/Fa#factorial">Index entries for sequences related to factorial numbers</a>
%H A346928 <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>
%e A346928 Triangle begins:
%e A346928     1;
%e A346928     2;
%e A346928     3, 6;
%e A346928     4, 12, 24;
%e A346928     5, 20, 60, 120;
%e A346928     30, 360, 720;
%e A346928     7, 42, 210, 840, 2520, 5040;
%e A346928     8, 56, 336, 1680, 6720, 20160, 40320;
%e A346928     9, 72, 504, 3024, 15120, 60480, 181440, 362880;
%e A346928     10, 90, 30240, 151200, 604800, 1814400, 3628800;
%e A346928     11, 110, 990, 7920, 55440, 332640, 1663200, 6652800, 19958400, 39916800;
%e A346928     ...
%o A346928 (PARI) s=[]; for (n=1, 11, p=1; forstep (m=n, 1, -1, if (!setsearch(s, p*=m), s=setunion(s, [p]); print1 (p", "))))
%o A346928 (Python)
%o A346928 from math import factorial
%o A346928 def auptor(rows):
%o A346928     alst, aset = [1], {1}
%o A346928     for n in range(2, rows+1):
%o A346928         fn = factorial(n)
%o A346928         for m in range(n-1, 0, -1):
%o A346928             fm = factorial(m)
%o A346928             q, r = divmod(fn, factorial(m))
%o A346928             if r == 0 and q not in aset:
%o A346928                 alst.append(q); aset.add(q)
%o A346928     return alst
%o A346928 print(auptor(11)) # _Michael S. Branicky_, Oct 17 2021
%Y A346928 Cf. A000142, A068424, A209260, A348401.
%K A346928 nonn,tabf
%O A346928 1,2
%A A346928 _Rémy Sigrist_, Oct 16 2021