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Number of nonzero elements in GF(2^n) that are 11th powers.
8

%I #31 Sep 08 2022 08:46:02

%S 1,3,7,15,31,63,127,255,511,93,2047,4095,8191,16383,32767,65535,

%T 131071,262143,524287,95325,2097151,4194303,8388607,16777215,33554431,

%U 67108863,134217727,268435455,536870911,97612893,2147483647,4294967295,8589934591,17179869183,34359738367,68719476735

%N Number of nonzero elements in GF(2^n) that are 11th powers.

%H Vincenzo Librandi, <a href="/A213247/b213247.txt">Table of n, a(n) for n = 1..1000</a>

%F a(n) = M / GCD( M, 11 ) where M=2^n-1.

%F From _Colin Barker_, Aug 24 2014: (Start)

%F a(n) = 1025*a(n-10)-1024*a(n-20).

%F G.f.: x*(512*x^18 +768*x^17 +896*x^16 +960*x^15 +992*x^14 +1008*x^13 +1016*x^12 +1020*x^11 +1022*x^10 +93*x^9 +511*x^8 +255*x^7 +127*x^6 +63*x^5 +31*x^4 +15*x^3 +7*x^2 +3*x +1) / (1024*x^20 -1025*x^10 +1).

%F (End)

%F a(n) = (2^n - 1)/11 if n is divisible by 10, 2^n - 1 otherwise. - _Robert Israel_, Aug 24 2014

%p A213247:=n->(2^n-1)/igcd(2^n-1,11): seq(A213247(n), n=1..40); # _Wesley Ivan Hurt_, Aug 24 2014

%t Table[(2^n - 1)/GCD[2^n - 1, 11], {n, 50}] (* _Vincenzo Librandi_, Mar 16 2013 *)

%o (Magma) [(2^n - 1) / GCD (2^n - 1, 11): n in [1..40]]; // _Vincenzo Librandi_, Mar 16 2013

%o (PARI) { for(n=1,36,if(n%10,a=2^n-1,a=(2^n-1)/11);print1(a,", ")) } \\ _K. Spage_, Aug 23 2014

%Y Cf. A213243 (cubes), A213244 (5th powers), A213245 (7th powers), A213246 (9th powers), A213248 (13th powers).

%K nonn

%O 1,2

%A _Joerg Arndt_, Jun 07 2012