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

1, 2, 3, 8, 15, 48, 105, 384, 945, 3840, 10395, 46080, 135135, 645120, 2027025, 3072, 155925, 256, 14175, 2816, 170100, 36608, 2381400, 549120, 11340, 32, 1260, 4, 6, 20, 36, 140, 288, 12600, 3168, 151200, 24, 5, 144, 35, 1152, 315, 16, 45, 112, 360, 14, 2880

Offset: 1

Author

Rémy Sigrist, Mar 14 2025

Keywords

• nonn

Comments

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!.

Examples

			The first terms are:
  n   a(n)     a(n)*a(n+1)
  --  -------  -----------
   1        1           2!
   2        2           3!
   3        3           4!
   4        8           5!
   5       15           6!
   6       48           7!
   7      105           8!
   8      384           9!
   9      945          10!
  10     3840          11!
  11    10395          12!
  12    46080          13!
  13   135135          14!
  14   645120          15!
  15  2027025          13!
  16     3072          12!
		

Links

• Rémy Sigrist, Table of n, a(n) for n = 1..10000
• Rémy Sigrist, PARI program

Crossrefs

Cf. A000142, A375579, A382072, A382083, A382085.

Programs

PARI

\\ See Links section.

Python

from itertools import count, islice
def agen(): # generator of terms
    fset, aset, an = set(), set(), 1
    while True:
        yield an
        aset.add(an)
        fk = 1
        for k in count(2):
            fk *= k
            q, r = divmod(fk, an)
            if r == 0 and q not in aset:
                an = q
                break
print(list(islice(agen(), 48))) # Michael S. Branicky, Mar 14 2025