A350176 Numbers m such that binomial(m, 3) divides binomial(3^m-2, 3).

3, 5, 7, 17, 79, 97, 257, 457, 65537, 677041, 1354081, 7812169, 13650001, 21381361, 65246161, 134246401, 242235841, 277032001, 393414001, 468930001, 793605121, 859560241, 886966081, 1609179001, 3355067041, 4269249601, 6024794161, 8120048641, 10142988241, 10466887201

Offset: 1

Author

Juri-Stepan Gerasimov, Dec 18 2021

Keywords

• nonn

Comments

Are all terms prime numbers?

Links

• Chai Wah Wu, Table of n, a(n) for n = 1..40

Crossrefs

Supersequence of A019434.

Cf. A069051 (binomial(k,2) divides binomial(2^k-1, 2)?).

Programs

Magma

[n: n in [3..10^4] |  IsZero(Binomial(3^n-2, 3) mod Binomial(n,3))];

Mathematica

Select[Range[3, 10^5], Divisible[Binomial[3^# - 2, 3], Binomial[#, 3]] &] (* Amiram Eldar, Dec 18 2021 *)

PARI

isok(m) = if (m>=3, (binomial(3^m-2, 3) % binomial(m, 3)) == 0); \\ Michel Marcus, Dec 19 2021

PARI

isok(m) = if (m>2, my(md = Mod(3, m^3 - 3*m^2 + 2*m)^m); (md^3 - 9*md^2 + 26*md - 24) == 0); \\ Michel Marcus, Dec 28 2021

Python

from itertools import count, islice
def A350176_gen(startvalue=3): # generator of terms >= startvalue
    for m in count(max(startvalue,3)):
        k = m*(m-1)*(m-2)
        a = pow(3,m,k)-2
        if (a*(a-1)*(a-2))%k == 0:
            yield m
A350176_list = list(islice(A350176_gen(),10)) # Chai Wah Wu, Jul 21 2025

Extensions

a(11)-a(20) from Michel Marcus, Dec 27 2021

a(21)-a(23) from Michel Marcus, Dec 28 2021

a(24)-a(30) from Chai Wah Wu, Jul 22 2025