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%I #14 Sep 08 2022 08:46:10
%S 1,20,26,32,54,162,204
%N Numbers k such that 7^k - k is a semiprime.
%C From _Robert Israel_, Sep 02 2016: (Start)
%C Odd k is in the sequence iff (7^k-k)/2 is prime.
%C If k == 1 (mod 3) then k is in the sequence iff (7^k-k)/3 is prime.
%C 708 is in the sequence but is not necessarily a(7). (End)
%C a(8) >= 384. - _Daniel Suteu_, Aug 05 2019
%e 1 is in this sequence because 7^1-1 = 2*3 is semiprime.
%e 20 is in this sequence because 7^20-20 = 1511201*52800564781 and these two factors are prime.
%p Res:= NULL:
%p for n from 1 to 100 do
%p F:= ifactors(7^n-n,easy)[2];
%p if add(t[2],t=F) >= 3 or (hastype(F,symbol) and add(t[2],t=F) >= 2)
%p then flag:= false
%p elif add(t[2],t=F) = 2 and not hastype(F,symbol) then flag:= true
%p else
%p flag:= evalb(numtheory:-bigomega(7^n-n)=2)
%p fi;
%p if flag then Res:= Res, n fi
%p od:
%p Res; # _Robert Israel_, Sep 02 2016
%t Select[Range[80], PrimeOmega[7^# - #]==2 &]
%o (Magma) IsSemiprime:=func<i | &+[d[2]: d in Factorization(i)] eq 2>; [m: m in [1..80] | IsSemiprime(s) where s is 7^m-m];
%Y Cf. similar sequences listed in A252656.
%K nonn,more
%O 1,2
%A _Vincenzo Librandi_, Dec 21 2014
%E a(6) from _Robert Israel_, Sep 02 2016
%E a(7) from _Daniel Suteu_, Aug 05 2019
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