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Search: id:A152049
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| A152049 |
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Number of conjugacy classes of primitive elements in GF(2^n) which have trace 0 |
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+0
1
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| 0, 0, 1, 1, 3, 2, 9, 9, 23, 29, 89, 72, 315, 375, 899, 1031, 3855, 3886
(list; graph; listen)
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OFFSET
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1,5
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COMMENT
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Also number of primitive polynomials of degree n over GF(2) whose second-highest coefficient is 0
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EXAMPLE
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a(3)=1 because of the two primitive degree 3 polynomials over GF(2), namely t^3+t+1 and t^3+t^2+1, only the former has a zero next-to-highest coefficient.
Similarly, a(13)=315, because of half (4096) of the 8192 elements of GF(2^13) have trace 0 and all except 0 (since 1 has trace 1) are primitive, so there are 4095/13=315 conjugacy classes of primitive elements of trace 0.
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PROGRAM
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(GAP) a := function(n) local q, k, cnt, x; q:=2^n; k:=GF(2, n); cnt:=0; for x in k do if Trace(k, GF(2), x)=0*Z(2) and Order(x)=q-1 then cnt := cnt+1; fi; od; return cnt/n; end;
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CROSSREFS
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Always less than A011260 (and exactly one half of it when 2^n-1 is prime)
Sequence in context: A081233 A050676 A010372 this_sequence A099887 A038220 A053151
Adjacent sequences: A152046 A152047 A152048 this_sequence A152050 A152051 A152052
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KEYWORD
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nonn
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AUTHOR
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David A. Madore (david.madore(AT)ens.fr), Nov 21 2008
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