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A027385 Number of primitive polynomials of degree n over GF(3).

%I A027385 #33 Feb 16 2025 08:32:35
%S A027385 1,2,4,8,22,48,156,320,1008,2640,7700,13824,61320,170352,401280,
%T A027385 983040,3796100,7838208,30566592,62304000,229686912,670824000,
%U A027385 2003046356,3583180800,15403487000,48881851200,128672022528,314657860608,1163185915872,2340264960000,9947788640064
%N A027385 Number of primitive polynomials of degree n over GF(3).
%C A027385 Second row of the array A158502(n, k) = phi(p^k-1)/k, p=prime(n). - _R. J. Mathar_, Aug 24 2011
%C A027385 From _Joerg Arndt_, Oct 03 2012: (Start)
%C A027385 Number of base-3, length-n Lyndon words w such that gcd(w, 3^n-1)==1 (where w is interpreted as a radix-3 number); replacing 3 by any prime p gives the analogous statement for GF(p).
%C A027385 The statement above is the consequence of the following.
%C A027385 Let p be a prime and g be a generator of GF(p^n). If w is a base-p, length-n Lyndon word then f=g^w (where w is interpreted as a radix-p number) has an irreducible characteristic polynomial C (over GF(p)) and, if gcd(w,p^n-1)==1 then C is primitive.
%C A027385 (End)
%H A027385 Vaclav Kotesovec, <a href="/A027385/b027385.txt">Table of n, a(n) for n = 1..200</a> (terms 1..100 from Seiichi Manyama)
%H A027385 Eric W. Weisstein, <a href="https://mathworld.wolfram.com/TotientFunction.html">MathWorld: Totient Function</a>
%H A027385 Wikipedia, <a href="http://en.wikipedia.org/wiki/Euler%27s_phi_function">Euler's totient function</a>
%p A027385 A027385 := proc(n) numtheory[phi](3^n-1)/n; end proc:
%t A027385 Table[EulerPhi[3^n - 1]/n, {n, 1, 30}] (* _Vaclav Kotesovec_, Nov 23 2017 *)
%o A027385 (PARI) a(n) = eulerphi(3^n-1)/n; /* _Joerg Arndt_, Aug 25 2011 */
%K A027385 nonn
%O A027385 1,2
%A A027385 _Frank Ruskey_