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A252659 Numbers m such that 6^m - m is a semiprime.

%I A252659 #17 Nov 12 2024 20:37:17
%S A252659 2,3,5,10,15,23,34,37,47,70,259,275,278,497,563
%N A252659 Numbers m such that 6^m - m is a semiprime.
%C A252659 From _Robert Israel_, Sep 06 2016: (Start)
%C A252659 Even n is in this sequence iff (6^n-n)/2 is prime.
%C A252659 3*k is in this sequence iff 2*6^(3*k-1)-k is prime.
%C A252659 Also contains 275, 278 and 683.
%C A252659 The only other possible member less than 275 is 259. (End)
%C A252659 a(16) >= 617. - _Kevin P. Thompson_, Apr 29 2022
%H A252659 FactorDB, <a href="http://factordb.com/index.php?id=1100000003337567816">Status of 6^617-617</a>
%e A252659 2 is in this sequence because 6^2-2 = 2*17 is semiprime.
%e A252659 10 is in this sequence because 6^10-10 = 2*30233083 and these two factors are prime.
%p A252659 Res:= NULL:
%p A252659 for n from 1 to 100 do
%p A252659     F:= ifactors(6^n-n, easy)[2];
%p A252659     if add(t[2], t=F) >= 3 or (hastype(F, symbol) and add(t[2], t=F) >= 2)
%p A252659        then flag:= false
%p A252659     elif add(t[2], t=F) = 2 and not hastype(F, symbol) then flag:= true
%p A252659     else
%p A252659      flag:= evalb(numtheory:-bigomega(6^n-n)=2)
%p A252659     fi;
%p A252659   if flag then  Res:= Res, n fi
%p A252659 od:
%p A252659 Res; # _Robert Israel_, Sep 06 2016
%t A252659 Select[Range[90], PrimeOmega[6^# - #]== 2&]
%o A252659 (Magma) IsSemiprime:=func<i | &+[d[2]: d in Factorization(i)] eq 2>; [m: m in [1..90] | IsSemiprime(s) where s is 6^m-m];
%Y A252659 Cf. similar sequences listed in A252656.
%K A252659 nonn,more,hard
%O A252659 1,1
%A A252659 _Vincenzo Librandi_, Dec 21 2014
%E A252659 a(11)-a(15) from _Kevin P. Thompson_, Apr 29 2022