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Sequence S such that 1 is in S and if x is in S, then 6x-1 and 6x+1 are in S.
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%I #19 Jan 30 2023 18:49:07

%S 1,5,7,29,31,41,43,173,175,185,187,245,247,257,259,1037,1039,1049,

%T 1051,1109,1111,1121,1123,1469,1471,1481,1483,1541,1543,1553,1555,

%U 6221,6223,6233,6235,6293,6295,6305,6307,6653,6655,6665,6667,6725,6727,6737,6739

%N Sequence S such that 1 is in S and if x is in S, then 6x-1 and 6x+1 are in S.

%C Does every generation after the initial 1 contain a prime?

%H Ivan Neretin, <a href="/A147993/b147993.txt">Table of n, a(n) for n = 1..10000</a>

%F Product_{j=0..n-1} cos(6^j) = 2^(-n+1)*Sum_{i=2^(n-1)..2^n-1} cos(a(i)). - _Gevorg Hmayakyan_, Jan 15 2017

%F Sum_{i=2^(n-1)..2^n-1} cos((a(i)/6^(n-1))*(Pi/2)) = 0. - _Gevorg Hmayakyan_, Jan 15 2017

%F a(n) = 2*A032806(n-1) + 1 for n>=2. - _Alois P. Heinz_, Jan 29 2023

%e 0th generation: 1

%e 1st generation: 5 7

%e 2nd generation: 29 31 41 43

%t sss[n_]:={6n-1,6n+1}; NestList[Sort[Flatten[sss[#]]]&,{1},5]//Flatten (* _Harvey P. Dale_, Aug 06 2018 *)

%Y Cf. A032806.

%Y Column k=6 of A360099.

%K nonn

%O 1,2

%A _Clark Kimberling_, Dec 07 2008