%I #40 Sep 08 2022 08:46:02

%S 1,1,7,5,31,7,127,85,511,341,2047,455,8191,5461,32767,21845,131071,

%T 29127,524287,349525,2097151,1398101,8388607,1864135,33554431,

%U 22369621,134217727,89478485,536870911,119304647,2147483647,1431655765,8589934591,5726623061,34359738367,7635497415

%N Number of nonzero elements in GF(2^n) that are 9th powers.

%H Vincenzo Librandi, <a href="/A213246/b213246.txt">Table of n, a(n) for n = 1..1000</a>

%H <a href="/index/Rec#order_12">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,0,0,0,65,0,0,0,0,0,-64).

%F a(n) = M / gcd( M, 9 ), where M=2^n-1.

%F Conjectures from _Colin Barker_, Aug 23 2014: (Start)

%F a(n) = 65*a(n-6)-64*a(n-12).

%F G.f.: x*(2*x^2 -x +1)*(16*x^8 +16*x^7 +28*x^6 +16*x^5 +25*x^4 +8*x^3 +7*x^2 +2*x +1) / ((x -1)*(x +1)*(2*x -1)*(2*x +1)*(x^2 -x +1)*(x^2 +x +1)*(4*x^2 -2*x +1)*(4*x^2 +2*x +1)). (End)

%F Conjectures verified by _Robert Israel_, Jun 27 2018.

%p A213246:=n->(2^n-1)/gcd(2^n-1,9): seq(A213246(n), n=1..40); # _Wesley Ivan Hurt_, Aug 24 2014

%t Table[(2^n - 1)/GCD[2^n - 1, 9], {n, 100}] (* _Vincenzo Librandi_, Mar 15 2013 *)

%o (Magma) [(2^n-1)/GCD(2^n-1, 9): n in [1..40]]; // _Vincenzo Librandi_, Mar 15 2013

%o (PARI) a(n)=(2^n-1)/gcd(2^n-1,9) \\ _Edward Jiang_, Sep 04 2014

%o (GAP) List([1..40],n->(2^n-1)/Gcd(2^n-1,9)); # _Muniru A Asiru_, Jun 27 2018

%Y Cf. A213243 (cubes), A213244 (5th powers), A213245 (7th powers), A213247 (11th powers), A213248 (13th powers).

%K nonn,easy

%O 1,3

%A _Joerg Arndt_, Jun 07 2012