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A057764 Triangle T(n,k) = number of nonzero elements of multiplicative order k in Galois field GF(2^n) (n >= 1, 1 <= k <= 2^n-1).

%I A057764 #19 Feb 16 2023 05:11:46
%S A057764 1,1,0,2,1,0,0,0,0,0,6,1,0,2,0,4,0,0,0,0,0,0,0,0,0,8,1,0,0,0,0,0,0,0,
%T A057764 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,30,1,0,2,0,0,0,6,0,6,0,0,
%U A057764 0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,36
%N A057764 Triangle T(n,k) = number of nonzero elements of multiplicative order k in Galois field GF(2^n) (n >= 1, 1 <= k <= 2^n-1).
%H A057764 Robert Israel, <a href="/A057764/b057764.txt">Table of n, a(n) for n = 1..16369</a> (rows 1 to 13, flattened)
%F A057764 From _Robert Israel_, Jul 21 2016: (Start)
%F A057764 T(n,k) = A000010(k) if k is a divisor of 2^n-1, otherwise 0.
%F A057764 Sum_{k=1..2^n-1} T(n,k) = 2^n-1 = A000225(n).
%F A057764 G.f. as triangle: g(x,y) = Sum_{j>=0} x^A002326(j)*A000010(2j+1)*y^(2j+1)/(1-x^A002326(j)). (End)
%e A057764 Table begins:
%e A057764   1;
%e A057764   1, 0, 2;
%e A057764   1, 0, 0, 0, 0, 0, 6;
%e A057764   ...
%p A057764 f:= proc(n,k) if 2^n-1 mod k = 0 then numtheory:-phi(k) else 0 fi end proc:
%p A057764 seq(seq(f(n,k),k=1..2^n-1), n=1..10); # _Robert Israel_, Jul 21 2016
%t A057764 T[n_, k_] := If[Divisible[2^n - 1, k], EulerPhi[k], 0];
%t A057764 Table[T[n, k], {n, 1, 10}, {k, 1, 2^n - 1}] // Flatten (* _Jean-François Alcover_, Feb 07 2023, after _Robert Israel_ *)
%o A057764 (Magma) {* Order(g) : g in GF(2^6) | g ne 0 *};
%Y A057764 Cf. A000010, A000225, A002326, A053287.
%K A057764 nonn,easy,nice,tabf
%O A057764 1,4
%A A057764 _N. J. A. Sloane_, Nov 01 2000
%E A057764 T(6,21) corrected by _Robert Israel_, Jul 21 2016