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

Author

Rémy Sigrist, Jun 24 2021

Keywords

• nonn

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.

Examples

			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.
		

Links

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

Crossrefs

Cf. A000142, A001710, A045619, A068424, A118235, A246655.

Programs

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

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.

a(n) < n iff n belongs to A045619.