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

%I A350176 #69 Aug 09 2025 15:15:03
%S A350176 3,5,7,17,79,97,257,457,65537,677041,1354081,7812169,13650001,
%T A350176 21381361,65246161,134246401,242235841,277032001,393414001,468930001,
%U A350176 793605121,859560241,886966081,1609179001,3355067041,4269249601,6024794161,8120048641,10142988241,10466887201
%N A350176 Numbers m such that binomial(m, 3) divides binomial(3^m-2, 3).
%C A350176 Are all terms prime numbers?
%H A350176 Chai Wah Wu, <a href="/A350176/b350176.txt">Table of n, a(n) for n = 1..40</a>
%t A350176 Select[Range[3, 10^5], Divisible[Binomial[3^# - 2, 3], Binomial[#, 3]] &] (* _Amiram Eldar_, Dec 18 2021 *)
%o A350176 (Magma) [n: n in [3..10^4] |  IsZero(Binomial(3^n-2, 3) mod Binomial(n,3))];
%o A350176 (PARI) isok(m) = if (m>=3, (binomial(3^m-2, 3) % binomial(m, 3)) == 0); \\ _Michel Marcus_, Dec 19 2021
%o A350176 (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
%o A350176 (Python)
%o A350176 from itertools import count, islice
%o A350176 def A350176_gen(startvalue=3): # generator of terms >= startvalue
%o A350176     for m in count(max(startvalue,3)):
%o A350176         k = m*(m-1)*(m-2)
%o A350176         a = pow(3,m,k)-2
%o A350176         if (a*(a-1)*(a-2))%k == 0:
%o A350176             yield m
%o A350176 A350176_list = list(islice(A350176_gen(),10)) # _Chai Wah Wu_, Jul 21 2025
%Y A350176 Supersequence of A019434.
%Y A350176 Cf. A069051 (binomial(k,2) divides binomial(2^k-1, 2)?).
%K A350176 nonn
%O A350176 1,1
%A A350176 _Juri-Stepan Gerasimov_, Dec 18 2021
%E A350176 a(11)-a(20) from _Michel Marcus_, Dec 27 2021
%E A350176 a(21)-a(23) from _Michel Marcus_, Dec 28 2021
%E A350176 a(24)-a(30) from _Chai Wah Wu_, Jul 22 2025