A375579 -id:A375579 - cp's OEIS frontend

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

Rémy Sigrist, Mar 14 2025

nonn

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!.

			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!

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

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

PARI
```
\\ See Links section.
```

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

A375615 Lexicographically earliest sequence of distinct positive integers such that for any n > 0, n * a(n) belongs to A055932.

Original entry on oeis.org

1, 2, 4, 3, 6, 5, 30, 8, 10, 9, 210, 12, 2310, 15, 14, 16, 30030, 18, 510510, 21, 20, 105, 9699690, 24, 36, 1155, 32, 45, 223092870, 7, 6469693230, 27, 70, 15015, 42, 25, 200560490130, 255255, 770, 48, 7420738134810, 35, 304250263527210, 315, 28, 4849845

Offset: 1

Rémy Sigrist, Aug 21 2024

nonn

This sequence is a self-inverse permutation of the positive integers.

The n-th term is a multiple of A083720(n).

			The first terms, alongside the prime factors of n*a(n), are:
  n   a(n)  Prime factors
  --  ----  --------------------
   1     1  {}
   2     2  {2}
   3     4  {2, 3}
   4     3  {2, 3}
   5     6  {2, 3, 5}
   6     5  {2, 3, 5}
   7    30  {2, 3, 5, 7}
   8     8  {2}
   9    10  {2, 3, 5}
  10     9  {2, 3, 5}
  11   210  {2, 3, 5, 7, 11}
  12    12  {2, 3}
  13  2310  {2, 3, 5, 7, 11, 13}
  14    15  {2, 3, 5, 7}
  15    14  {2, 3, 5, 7}

Rémy Sigrist, PARI program
Index entries for sequences that are permutations of the natural numbers

Cf. A055932, A083720, A375579.

PARI
```
\\ See Links section.
```

cp's OEIS Frontend

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

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