A320920 a(n) is the smallest number m such that binomial(m,n) is nonzero and is divisible by n!.

1, 4, 9, 33, 28, 165, 54, 1029, 40832, 31752, 28680, 2588680, 2162700, 12996613, 12341252, 4516741125, 500367376, 133207162881, 93770874890, 7043274506259, 40985291653137, 70766492123145, 321901427163142, 58731756479578128, 676814631896875010, 6820060161969750025

Offset: 1

Author

Tanya Khovanova, Oct 24 2018

Keywords

• nonn

Comments

a(n) is such that a nontrivial n-symmetric permutation of [1..a(n)] might exist.

Examples

			The sequence of binomial coefficients C(n,3) starts as: 0, 0, 1, 4, 10, 20, 35, 56, 84, 120, 165, and so on. The smallest nonzero number divisible by 3! is 84, which is C(9,3). Therefore a(3) = 9.
		

Links

• Bert Dobbelaere, Table of n, a(n) for n = 1..34
• Bert Dobbelaere, Python program
• Tanya Khovanova, 3-Symmetric Permutations

Crossrefs

Cf. A042948, A316775, A320919.

Programs

Mathematica

a[n_] := Module[{w, m, bc}, {w, m} = {n!, n}; bc[i_] := Binomial[n-1, i] ~Mod~ w; While[True, bc[n] = (bc[n-1] + bc[n]) ~Mod~ w; If[bc[n] == 0, Return[m]]; For[i = n-1, i >= 0, i--, bc[i] = (bc[i-1] + bc[i]) ~Mod~ w]; m++]];
Array[a, 12] (* Jean-François Alcover, May 31 2019, after Chai Wah Wu *)

Python

from sympy import factorial, binomial
def A320920(n):
    w, m = int(factorial(n)), n
    bc = [int(binomial(n-1,i)) % w for i in range(n+1)]
    while True:
        bc[n] = (bc[n-1]+bc[n]) % w
        if bc[n] == 0:
            return m
        for i in range(n-1,0,-1):
            bc[i] = (bc[i-1]+bc[i]) % w
        m += 1 # Chai Wah Wu, Oct 25 2018

Extensions

a(14)-a(15) from Alois P. Heinz, Oct 24 2018

a(16)-a(17) from Chai Wah Wu, Oct 25 2018

a(18)-a(19) from Giovanni Resta, Oct 26 2018

a(20) from Giovanni Resta, Oct 27 2018

a(21) and beyond from Bert Dobbelaere, Feb 11 2020