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A213245 Number of nonzero elements in GF(2^n) that are 7th powers. 8

%I #36 Sep 08 2022 08:46:02

%S 1,3,1,15,31,9,127,255,73,1023,2047,585,8191,16383,4681,65535,131071,

%T 37449,524287,1048575,299593,4194303,8388607,2396745,33554431,

%U 67108863,19173961,268435455,536870911,153391689,2147483647,4294967295,1227133513,17179869183,34359738367,9817068105

%N Number of nonzero elements in GF(2^n) that are 7th powers.

%H Vincenzo Librandi, <a href="/A213245/b213245.txt">Table of n, a(n) for n = 1..1000</a>

%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,9,0,0,-8).

%F a(n) = M / gcd( M, 7 ), where M=2^n-1.

%F Conjectures from _Colin Barker_, Aug 23 2014, verified by _Robert Israel_, Nov 20 2016: (Start)

%F a(n) = 9*a(n-3)-8*a(n-6).

%F G.f.: x*(4*x^4+6*x^3+x^2+3*x+1) / ( (x-1)*(2*x-1)*(x^2+x+1)*(4*x^2+2*x+1) ). (End)

%p A213245:=n->(2^n-1)/gcd(2^n-1,7): seq(A213245(n), n=1..40); # _Wesley Ivan Hurt_, Aug 24 2014

%t Table[(2^n - 1)/GCD[2^n - 1, 7], {n, 60}] (* _Vincenzo Librandi_, Mar 16 2013 *)

%o (Magma) [(2^n - 1) / GCD (2^n - 1, 7): n in [1..40]]; // _Vincenzo Librandi_, Mar 16 2013

%o (PARI) a(n)=(2^n-1)/gcd(2^n-1,7) \\ _Edward Jiang_, Sep 04 2014

%Y Cf. A213243 (cubes), A213244 (5th powers), A213246 (9th powers), A213247 (11th powers), A213248 (13th powers).

%K nonn,easy

%O 1,2

%A _Joerg Arndt_, Jun 07 2012

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Last modified March 28 16:34 EDT 2024. Contains 371254 sequences. (Running on oeis4.)