A353010 a(n) = maximal d such that Product_{k=0..m} binomial(m,k) is divisible by m^(m+d), where m = A276710(n).

0, 0, 3, 0, 1, 9, 49, 0, 21, 19, 31, 73, 0, 61, 57, 16, 4, 46, 13, 43, 25, 0, 20, 106, 1, 57, 172, 81, 43, 66, 25, 29, 51, 41, 38, 140, 80, 1, 71, 0, 0, 34, 117, 59, 199, 134, 208, 181, 9, 55, 259, 202, 114, 28, 263, 100, 145, 32, 157, 217, 60, 121, 36, 73, 86, 94, 19, 67, 154, 21, 40, 73, 57, 167, 392, 135, 256

Offset: 1

Author

Hagen von Eitzen, Apr 15 2022

Keywords

• nonn

Comments

By definition of A276710, a(n) >= -1.

It is conjectured that a(n) >= 0, computationally verified up to n = 10^7.

Empirically from terms up to n=10^7, a(n) seems to become quite large, small values are rare, and yet a(n)=0 also seems to occur for large n.

Examples

			The 7th term of A276710 is 105 because Product_{k=1..105} binomial(36,k) is divisible by 105^(105-1). Actually, it is divisible by 105^(105+49), but not by 105^(105+50). Therefore, a(7) = 49.
		

Crossrefs

Cf. A276710.

Programs

Python

from math import prod, comb
from itertools import islice
from sympy import nextprime
def A353010_gen(): # generator of terms
    p, q = 3, 5
    while True:
        for m in range(p+1,q):
            r = m**(m-1)
            c = 1
            for k in range(m+1):
                c = c*comb(m,k) % r
            if c == 0:
                d, (e, f) = -m, divmod(prod(comb(m,k) for k in range(m+1)),m)
                while f == 0:
                    d += 1
                    e, f = divmod(e,m)
                yield d
        p, q = q, nextprime(q)
A353010_list = list(islice(A353010_gen(),40)) # Chai Wah Wu, Jun 09 2022