login
Numbers k such that binomial(k, 3) divides binomial(3^k-3, 2).
0

%I #11 Mar 09 2022 00:18:09

%S 3,5,17,37,79,101,257,14401,44101,47881,57601,65537,677041,1354081,

%T 2031121,3766141,7812169,8122501,17907121,18671941,19676161,21381361,

%U 29615041,30115009,65246161,105557761,144406081,181254529,217039681,242235841,263062801,277032001

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

%C Are all terms prime numbers?

%C Conjecture: all terms of the intersection with A350176 are prime numbers.

%t Select[Range[3, 70000], Divisible[Binomial[3^# - 3, 2], Binomial[#, 3]] &] (* _Amiram Eldar_, Jan 21 2022 *)

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

%o (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 Supersequence of A019434.

%Y Cf. A350176, A350402.

%K nonn

%O 1,1

%A _Juri-Stepan Gerasimov_, Jan 21 2022

%E More terms from _Jinyuan Wang_, Jan 22 2022