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A350905 Numbers k such that binomial(k, 3) divides binomial(3^k-3, 2).

%I A350905 #15 Aug 22 2025 20:17:48
%S A350905 3,5,17,37,79,101,257,14401,44101,47881,57601,65537,677041,1354081,
%T A350905 2031121,3766141,7812169,8122501,17907121,18671941,19676161,21381361,
%U A350905 29615041,30115009,65246161,105557761,144406081,181254529,217039681,242235841,263062801,277032001
%N A350905 Numbers k such that binomial(k, 3) divides binomial(3^k-3, 2).
%C A350905 Are all terms prime numbers?
%C A350905 Conjecture: all terms of the intersection with A350176 are prime numbers.
%H A350905 Chai Wah Wu, <a href="/A350905/b350905.txt">Table of n, a(n) for n = 1..86</a>
%t A350905 Select[Range[3, 70000], Divisible[Binomial[3^# - 3, 2], Binomial[#, 3]] &] (* _Amiram Eldar_, Jan 21 2022 *)
%o A350905 (Magma) [n: n in [3..10^4] |  IsZero(Binomial(3^n-3, 2) mod Binomial(n, 3))];
%o A350905 (PARI) is(k) = if(k>2, my(m=Mod(3, (k^3+2*k)/3-k^2)^k);  m^2-7*m+12==0); \\ _Jinyuan Wang_, Jan 22 2022
%Y A350905 Supersequence of A019434.
%Y A350905 Cf. A350176, A350402.
%K A350905 nonn,changed
%O A350905 1,1
%A A350905 _Juri-Stepan Gerasimov_, Jan 21 2022
%E A350905 More terms from _Jinyuan Wang_, Jan 22 2022