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COMMENTS
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Second row of the array T(n,k) = phi(p^k-1)/k, p=prime(n), which starts
1, 1, 2, 2, 6, 6, 18, 16, ... A011260
1, 2, 4, 8, 22, 48, 156, 320, ... here
2, 4, 20, 48, 280, 720, 5580, 14976, ... A027741
2, 8, 36, 160, 1120, 6048, 37856, 192000, ... A027743
4, 16, 144, 960, 12880, 62208, 1087632, 7027200, ... A319166
4, 24, 240, 1536, 24752, 224640, 2988024, 21934080, ...
8, 48, 816, 5376, 141984, 1057536, 29309904, 224501760, ...
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).
The statement above is the consequence of the following.
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.
(End)
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