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A073726
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Primitive irreducible trinomials: x^n + x^k + 1 is a primitive irreducible polynomial (mod 2) for some k with 0 < k < n.
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8
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2, 3, 4, 5, 6, 7, 9, 10, 11, 15, 17, 18, 20, 21, 22, 23, 25, 28, 29, 31, 33, 35, 36, 39, 41, 47, 49, 52, 55, 57, 58, 60, 63, 65, 68, 71, 73, 79, 81, 84, 87, 89, 93, 94, 95, 97, 98, 100, 103, 105, 106, 108, 111, 113, 118, 119, 121, 123, 124, 127, 129, 130, 132, 134, 135, 137, 140, 142, 145, 148, 150, 151, 153, 159, 161, 167, 169, 170, 172, 174, 175, 177, 178, 183, 185, 191, 193, 194, 198, 199, 201
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,1
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COMMENTS
| Starts similar to A194125, first terms missing there are 140, 212, 236.
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REFERENCES
| S. W. Golomb, "Shift Register Sequences", revised edition, reprinted by Aegean Park Press, 1982. See Tables V-1, V-2.
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LINKS
| Joerg Arndt, Complete list of primitive trinomials over GF(2) up to degree 400.
A. J. Menezes, P. C. van Oorschot and S. A. Vanstone, Handbook of Applied Cryptography, CRC Press, 1996; see Table 4.8.
Index entries for sequences related to trinomials over GF(2)
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MAPLE
| A073726 := proc(n) local k, m: option remember: if(n=1)then return 2: else m:=procname(n-1)+1: while(true)do for k from 1 to m-1 do if Primitive(x^m+x^k+1) mod 2 then return m: fi: od: m:=m+1: od: fi: end:
seq(A073726(n), n=1..20); # Nathaniel Johnston, Apr 26 2011
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PROG
| (MAGMA) A073726 := function(n) for k := 1 to n-1 do if IsPrimitive(x^n+x^k+1) then return true; end if; end for; return false; end function; l := []; for n := 1 to 100 do if A073726(n) then l := Append(l, n); end if; end for; l;
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CROSSREFS
| See A073726 for irreducible trinomials and A001153 for primitive Mersenne trinomials (and references). See A074744 for values of k.
Cf. A194125 (n such that x^n+(1+x)^w over GF(2) is primitive for some w).
Sequence in context: A071557 A179401 A117729 * A194125 A008839 A039226
Adjacent sequences: A073723 A073724 A073725 * A073727 A073728 A073729
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KEYWORD
| nonn,nice
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AUTHOR
| Richard Brent and Paul Zimmermann, Sep 05 2002
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EXTENSIONS
| a(49) - a(58) from Nathaniel Johnston (nathaniel(AT)nathanieljohnston.com), Apr 26 2011
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