

A057486


Degrees of absolutely reducible trinomials, i.e., numbers n such that x^n + x^m + 1 is factorable for all m between 1 and n.


4



8, 13, 16, 19, 24, 26, 27, 32, 37, 38, 40, 43, 45, 48, 50, 51, 53, 56, 59, 61, 64, 67, 69, 70, 72, 75, 77, 78, 80, 82, 83, 85, 88, 91, 96, 99, 101, 104, 107, 109, 112, 114, 115, 116, 117, 120, 122, 125, 128, 131, 133, 136, 138, 139, 141, 143, 144, 149, 152, 157
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OFFSET

1,1


COMMENTS

Brent, Hart, Kruppa, and Zimmermann found that 57885161 is a member of this sequence.  Charles R Greathouse IV, May 30 2013


LINKS

T. D. Noe and Charles R Greathouse IV, Table of n, a(n) for n = 1..5000 (first 200 terms from T. D. Noe)
Richard Brent, The Software gf2x
Paul Zimmermann, There is no primitive trinomial of degree 57885161 over GF(2), posting to NMBRTHRY mailing list [alternate link]
Index entries for sequences related to trinomials over GF(2)


EXAMPLE

a(1) = 8 because x^8 + x^1 + 1 = (1 + x + x^2)*(1 + x^2 + x^3 + x^5 + x^6) and x^8 + x^2 + 1 = (1 + x + x^4)^2 and x^8 + x^3 + 1 = (1 + x + x^3)*(1 + x + x^2 + x^3 + x^5) and x^8 + x^4 + 1 = (1 + x + x^2)^4 and x^8 + x^5 + 1 = (1 + x^2 + x^3)*(1 + x^2 + x^3 + x^4 + x^5) and x^8 + x^6 + 1 = (1 + x^3 + x^4)^2 and x^8 + x^7 + 1 = (1 + x + x^2)*(1 + x + x^3 + x^4 + x^6).


MATHEMATICA

Do[ k = 1; While[ ToString[ Factor[ x^n + x^k + 1, Modulus > 2 ]] != ToString[ x^n + x^k + 1 ] && k < n, k++ ]; If[ k == n, Print[ n ]], {n, 2, 234} ]


PROG

(PARI) is(n)=for(s=1, n\2, if(polisirreducible((x^n+x^s+1)*Mod(1, 2)), return(0))); 1 \\ Charles R Greathouse IV, May 30 2013


CROSSREFS

Complement of A073571. Cf. A001153, A002475, A073639.
Sequence in context: A054295 A287159 A272486 * A188198 A129410 A168339
Adjacent sequences: A057483 A057484 A057485 * A057487 A057488 A057489


KEYWORD

nonn


AUTHOR

Robert G. Wilson v, Sep 28 2000


STATUS

approved



