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A213248 Number of nonzero elements in GF(2^n) that are 13th powers.

%I A213248 #35 Jan 11 2025 10:57:34
%S A213248 1,3,7,15,31,63,127,255,511,1023,2047,315,8191,16383,32767,65535,
%T A213248 131071,262143,524287,1048575,2097151,4194303,8388607,1290555,
%U A213248 33554431,67108863,134217727,268435455,536870911,1073741823,2147483647,4294967295,8589934591,17179869183,34359738367,5286113595
%N A213248 Number of nonzero elements in GF(2^n) that are 13th powers.
%H A213248 Vincenzo Librandi, <a href="/A213248/b213248.txt">Table of n, a(n) for n = 1..1000</a>
%H A213248 <a href="/index/Rec#order_24">Index entries for linear recurrences with constant coefficients</a>, signature (0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4097, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -4096).
%F A213248 a(n) = M / gcd( M, 13 ) where M=2^n-1.
%F A213248 Conjectures from _Colin Barker_, Aug 24 2014: (Start)
%F A213248 a(n) = 4097*a(n-12)-4096*a(n-24).
%F A213248 G.f.: x*(2048*x^22 +3072*x^21 +3584*x^20 +3840*x^19 +3968*x^18 +4032*x^17 +4064*x^16 +4080*x^15 +4088*x^14 +4092*x^13 +4094*x^12 +315*x^11 +2047*x^10 +1023*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) / (4096*x^24 -4097*x^12 +1). (End)
%p A213248 A213248:=n->(2^n-1)/gcd(2^n-1,13): seq(A213248(n), n=1..40); # _Wesley Ivan Hurt_, Aug 24 2014
%t A213248 Table[(2^n - 1)/GCD[2^n - 1, 13], {n, 40}] (* _Vincenzo Librandi_, Mar 17 2013 *)
%o A213248 (Magma) [(2^n - 1) / GCD (2^n - 1, 13): n in [1..40]]; // _Vincenzo Librandi_, Mar 17 2013
%o A213248 (PARI) a(n)=(2^n-1)/gcd(2^n-1,13) \\ _Edward Jiang_, Sep 04 2014
%Y A213248 Cf. A213243 (cubes), A213244 (5th powers), A213245 (7th powers), A213246 (9th powers), A213247 (11th powers).
%K A213248 nonn
%O A213248 1,2
%A A213248 _Joerg Arndt_, Jun 07 2012