A345708 a(n) is the least positive number starting an interval of consecutive integers whose product of elements is n.

1, 1, 3, 4, 5, 1, 7, 8, 9, 10, 11, 3, 13, 14, 15, 16, 17, 18, 19, 4, 21, 22, 23, 1, 25, 26, 27, 28, 29, 5, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 6, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 7, 57, 58, 59, 3, 61, 62, 63, 64, 65, 66, 67, 68, 69

Offset

1,3

Comments

This sequence is similar to A118235; here we multiply, there we add.

a(n) is the index of the first row of A068424 (interpreted as a square array) containing n.

If n is the product of k consecutive integers, then k! divides n.

Links

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

Formula

a(n) = 1 iff n is a factorial number (A000142).

a(n) <> 2.

a(n) = 3 iff n >= 3 and n belongs to A001710.

a(n) <= n.

a(p! / (n-1)!) = n for any n >= 3 and any prime number p >= n.

a(q) = q for any prime power q > 2.

a(n) = n for any odd number n.

Example

The square array A068424(n, k) begins:
  n\k|   1    2     3      4       5        6
  ---+---------------------------------------
    1|   1    2     6     24     120      720
    2|   2    6    24    120     720     5040
    3|   3   12    60    360    2520    20160
    4|   4   20   120    840    6720    60480
- so a(1) = a(2) = a(6) = a(24) = a(120) = a(720) = 1,
     a(3) = a(12) = a(60) = a(360) = 3,
     a(4) = a(20) = 4.

Prog

(PARI) a(n) = { fordiv (n, d, my (r=n); for (k=d, oo, if (r==1, return (d), r%k, break, r/=k))) }
(PARI) a(n) = { for (i=2, oo, if (n%i!, forstep (j=i-1, 2, -1, my (d=sqrtnint(n, j)); while (d-1 && n%(d-1)==0, d--); while (n%d==0, my (r=n); for
(k=d, oo, if (r==1, return (if (d==2, 1, d)), r%k, break, r/=k)); d++)); break)); return (n) }
(Python)
from sympy import divisors
def a(n):
    if n%2 == 0: return n
    divs = divisors(n)
    for i, d in enumerate(divs[:len(divs)//2]):
        p = lastj = d
        for j in divs[i+1:]:
            if p >= n or j - lastj > 1: break
            p, lastj = p*j, j
        if p == n: return d
    return n
print([a(n) for n in range(1, 70)]) # Michael S. Branicky, Jun 29 2021

Crossrefs

Cf. A000142, A001710, A068424, A118235, A246655.

Sequence in context: A350091 A135347 A370116 * A144786 A247369 A009389

Adjacent sequences: A345705 A345706 A345707 * A345709 A345710 A345711

Keyword

nonn

Author

Rémy Sigrist, Jun 24 2021

Status

approved