A249430 a(n) = Least integer k such that A249431(k) = n, and -1 if no such integer exists.

1, 0, 350, 439, 174, 713, 323, 1923, 1052, 999, 1766, 3749, 2254, 2253, 1934, 3391, 4184, 4463, 3144, 5451, 9698, 16279, 6398, 5123, 2974, 12863, 19094, 4299, 16574, 5749

Offset: 0

Author

Antti Karttunen, Nov 02 2014

Keywords

• nonn
• more
• changed

Comments

a(n) = the least natural number k such that {product of elements on row k of Pascal's triangle} is divisible by (k+n)! but not by (k+n+1)!

Note: a(18) = 3144 and a(24) = 2974. First values k for which A249431(k) = 16 and 17, if they exist, are larger than 4096.

Crossrefs

Nonnegative terms are all members of A249434.

Cf. A000142, A001142, A007318, A249151, A249431, A249432.

Programs

Python

from itertools import count
from math import factorial
def A249430(n):
    f = factorial(n)
    g = f*(n+1)
    pascal = [1]
    for k in count(0):
        a = 1
        for i in range(k+1):
            a = a*pascal[i]%f
        if not a:
            b = 1
            for i in range(k+1):
                b = b*pascal[i]%g
            if b:
                return k
        f = g
        g *= k+n+2
        pascal = [1]+[pascal[i]+pascal[i+1] for i in range(k)]+[1] # Chai Wah Wu, Aug 18 2025

Scheme

(define (A249430 n) (let loop ((k 0)) (cond ((= n (A249431 k)) k) (else (loop (+ 1 k))))))

Extensions

a(16)-a(20) from Chai Wah Wu, Aug 19 2025

a(21)-a(29) from Chai Wah Wu, Aug 27 2025