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A382067 Lexicographically earliest sequence of distinct positive integers such that the product of two consecutive terms is always a factorial number.

%I A382067 #12 Mar 17 2025 22:19:57
%S A382067 1,2,3,8,15,48,105,384,945,3840,10395,46080,135135,645120,2027025,
%T A382067 3072,155925,256,14175,2816,170100,36608,2381400,549120,11340,32,1260,
%U A382067 4,6,20,36,140,288,12600,3168,151200,24,5,144,35,1152,315,16,45,112,360,14,2880
%N A382067 Lexicographically earliest sequence of distinct positive integers such that the product of two consecutive terms is always a factorial number.
%C A382067 For any prime number p, the sequence contains a multiple of p, say a(k), and this term satisfies a(k-1)*a(k) = p!.
%H A382067 Rémy Sigrist, <a href="/A382067/b382067.txt">Table of n, a(n) for n = 1..10000</a>
%H A382067 Rémy Sigrist, <a href="/A382067/a382067.gp.txt">PARI program</a>
%e A382067 The first terms are:
%e A382067   n   a(n)     a(n)*a(n+1)
%e A382067   --  -------  -----------
%e A382067    1        1           2!
%e A382067    2        2           3!
%e A382067    3        3           4!
%e A382067    4        8           5!
%e A382067    5       15           6!
%e A382067    6       48           7!
%e A382067    7      105           8!
%e A382067    8      384           9!
%e A382067    9      945          10!
%e A382067   10     3840          11!
%e A382067   11    10395          12!
%e A382067   12    46080          13!
%e A382067   13   135135          14!
%e A382067   14   645120          15!
%e A382067   15  2027025          13!
%e A382067   16     3072          12!
%o A382067 (Python)
%o A382067 from itertools import count, islice
%o A382067 def agen(): # generator of terms
%o A382067     fset, aset, an = set(), set(), 1
%o A382067     while True:
%o A382067         yield an
%o A382067         aset.add(an)
%o A382067         fk = 1
%o A382067         for k in count(2):
%o A382067             fk *= k
%o A382067             q, r = divmod(fk, an)
%o A382067             if r == 0 and q not in aset:
%o A382067                 an = q
%o A382067                 break
%o A382067 print(list(islice(agen(), 48))) # _Michael S. Branicky_, Mar 14 2025
%o A382067 (PARI) \\ See Links section.
%Y A382067 Cf. A000142, A375579, A382072, A382083, A382085.
%K A382067 nonn
%O A382067 1,2
%A A382067 _Rémy Sigrist_, Mar 14 2025