The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation. Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A002475 Numbers k such that x^k + x + 1 is irreducible over GF(2). (Formerly M0544 N0194) 23
 0, 2, 3, 4, 6, 7, 9, 15, 22, 28, 30, 46, 60, 63, 127, 153, 172, 303, 471, 532, 865, 900, 1366, 2380, 3310, 4495, 6321, 7447, 10198, 11425, 21846, 24369, 27286, 28713, 32767, 34353, 46383, 53484, 62481, 83406, 87382, 103468, 198958 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS k=1 is excluded since the polynomial "1" is not normally regarded as irreducible. 2^(A073639(m)) - 1 is a term for all m. - Joerg Arndt, Aug 23 2015 REFERENCES N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence). N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence). S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 975. LINKS Joerg Arndt, Matters Computational (The Fxtbook), section 40.9.3 "Irreducible trinomials of the form 1 + x^k + x^d", p.850 N. Zierler, On x^n+x+1 over GF(2), Information and Control, 16 1970 502-505. MAPLE select(n -> Irreduc(x^n+x+1) mod 2, [0, \$2..10000]); # Robert Israel, Aug 09 2015 MATHEMATICA Do[ If[ ToString[ Factor[ x^n + x + 1, Modulus -> 2 ] ] == ToString[ x^n + x + 1 ], Print [ n ] ], {n, 0, 28713} ] Select[Range, IrreduciblePolynomialQ[x^# + x + 1, Modulus -> 2] &] (* Robert Price, Sep 19 2018 *) PROG (MAGMA) P := PolynomialRing(GaloisField(2)); for n := 2 to 100000 do if IsIrreducible(x^n+x+1) then print(n); end if; endfor; (Sage) P. = GF(2)[] for n in range(90):        if (x^n+x+1).is_irreducible():            print(n) # Ruperto Corso, Dec 11 2011 (PARI) for (n=1, 10^6, if ( polisirreducible(Mod(1, 2)*(x^n+x+1)), print1(n, ", ") ) ); /* Joerg Arndt, Apr 28 2012 */ (PARI) is(n)=if(n>3&&[1, 0, 1, 1, 0, 1, 0, 0][n%8+1], return(0)); polisirreducible(Mod('x^n+'x+1, 2)) \\ Charles R Greathouse IV, Jun 04 2015 CROSSREFS Cf. A001153, A073639, A057496, A223938 (n such that x^n-x-1 is irreducible over GF(3)). Sequence in context: A165773 A064414 A224482 * A208281 A306074 A250252 Adjacent sequences:  A002472 A002473 A002474 * A002476 A002477 A002478 KEYWORD nonn,hard,more,nice AUTHOR EXTENSIONS Two more terms from Paul Zimmermann, Sep 05 2002 a(37)-a(39) from Max Alekseyev, Oct 29 2011 a(40)-a(41) from Ruperto Corso, Dec 11 2011 a(42) from Manfred Scheucher, Jun 04 2015 a(43) from Manfred Scheucher, Aug 09 2015 STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified September 24 15:33 EDT 2021. Contains 347643 sequences. (Running on oeis4.)