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%I #19 Sep 08 2022 08:46:04
%S 2,3,4,5,9,13,41,121
%N Numbers n such that n, 2n-1 and 2n+1 all are prime or a prime power.
%C a(9) = (3^541-1)/2. - Conjectured by _Jack Brennen_, Oct 01 2012; confirmed by _Max Alekseyev_, Nov 10 2019
%C Since one of n, 2n-1, 2n+1 is divisible by 3 and thus is a power of 3, every term has one of the forms: 3^k, (3^k-1)/2, or (3^k+1)/2. - _Max Alekseyev_, Nov 10 2019
%H Max Alekseyev, <a href="/A217376/b217376.txt">Table of n, a(n) for n = 1..9</a>
%H W. Smith and others, <a href="http://groups.yahoo.com/group/primenumbers/message/24505">n, 2n-1, 2n+1 all prime or prime-power (maybe n-2 also)</a>, on primenumbers Yahoo! group, Oct 01 2012.
%H Warren Smith, Jack Brennen, Phil Carmody, <a href="/A217376/a217376.txt">n, 2n-1, 2n+1 all prime or prime-power (maybe n-2 also)</a>, digest of 6 messages in primenumbers Yahoo group, Oct 1, 2012.
%t Select[Range[200],Length[FactorInteger[#]]==Length[FactorInteger[2#-1]] == Length[FactorInteger[2#+1]]==1&] (* _Harvey P. Dale_, Oct 12 2012 *)
%o (PARI) for(n=1,9e9,omega(n)==1 & omega(2*n-1)==1 & omega(2*n+1)==1 & print1(n",")) \\ - _M. F. Hasler_, Oct 01 2012
%o (Magma) [k:k in [2..1000]| forall{s:s in [k,2*k-1,2*k+1]| #PrimeDivisors(s) eq 1}]; // _Marius A. Burtea_, Nov 13 2019
%K nonn
%O 1,1
%A _M. F. Hasler_, after an idea from Warren Smith, Oct 01 2012
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