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

1,2

Comments

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

Links

Bert Dobbelaere, Table of n, a(n) for n = 1..34

Bert Dobbelaere, Python program

Tanya Khovanova, 3-Symmetric Permutations

Example

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.

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 *)

Prog

(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

Crossrefs

Cf. A042948, A316775, A320919.

Sequence in context: A129196 A119574 A006393 * A368683 A048757 A356825

Adjacent sequences: A320917 A320918 A320919 * A320921 A320922 A320923

Keyword

nonn

Author

Tanya Khovanova, Oct 24 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

Status

approved