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A107220
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Numbers n such that 1 + (x + x^3 + x^5 + x^7 + ...+ x^(2*n+1)) is irreducible over GF(2).
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0
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1, 3, 5, 7, 9, 13, 23, 27, 31, 37, 63, 69, 117, 119, 173, 219, 223, 247, 307, 363, 383, 495, 695, 987, 2519, 3919, 4633, 6537, 8881, 12841, 15935, 16383, 16519, 26525, 34415, 95139
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OFFSET
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1,2
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COMMENTS
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All terms are odd as irreducible polynomials over GF(2) necessarily have an odd number of nonzero coefficients.
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LINKS
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EXAMPLE
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The number 5 is in the sequence because x^11 + x^9 + x^7 + x^5 + x^3 + x + 1 is irreducible over GF(2) (and 11 = 2*5 + 1).
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PROG
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(PARI) forstep(d=1, 10^5, 2, p=(1+sum(t=0, d, x^(2*t+1))); if(polisirreducible(Mod(1, 2)*p), print1(d, ", ")));
(Sage)
p = 1;
P.<x> = GF(2)[]
for n in range(1, 10^5, 2):
p = p + x^(2*(n-1)+1) + x^(2*n+1);
if p.is_irreducible():
print(n)
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CROSSREFS
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KEYWORD
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nonn,hard,more
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AUTHOR
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EXTENSIONS
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More terms from Joerg Arndt, Apr 02 2011 and (terms >=2519), Apr 27 2012
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STATUS
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approved
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