|
%I #17 Oct 30 2015 16:51:45
%S 9,243,289,9801,332929,11309769,384199201,13051463049,443365544449,
%T 15061377048201,511643454094369,17380816062160329,590436102659356801,
%U 20057446674355970889,681362750825443653409
%N Safe perfect powers: perfect powers n such that (n-1)/2 is also a perfect power.
%C If both powers are squares, the smaller square is a triangular number, and all square triangular numbers (A001110) correspond to a member in this sequence. This proves that this sequence is infinite. Are there only finitely many other members, i.e., is A075127 \ A055792 finite? - _Charles R Greathouse IV_, Dec 12 2010
%F Conjectures from _Colin Barker_, Oct 28 2015: (Start)
%F a(n) = 35*a(n-1)-35*a(n-2)+a(n-3) for n>5.
%F G.f.: x*(234*x^4-8182*x^3+7901*x^2+72*x-9) / ((x-1)*(x^2-34*x+1)).
%F (End)
%t pp = Select[ Range[10^8], Apply[ GCD, Last[ Transpose[ FactorInteger[ # ]]]] > 1 & ]; Select[pp, Apply[GCD, Last[ Transpose[ FactorInteger[( # - 1)/2]]]] > 1 & ]
%o (PARI) for(n=1, 1e10, if(ispower(n) && ispower((n-1)/2), print1(n, ", "))) \\ _Altug Alkan_, Oct 28 2015
%Y Cf. A001110, A001597, A055792, A070428, A075114.
%K nonn
%O 1,1
%A _Zak Seidov_, Oct 11 2002
%E One more term from _Robert G. Wilson v_, Oct 16 2002
%E a(7)-a(15) from _Donovan Johnson_, Mar 10 2010
|
