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A057646 a(n) = number of trinomials x^n + x^k + 1 that are irreducible over GF(2) for some k with n > k > 0. 4
1, 2, 2, 2, 3, 4, 0, 4, 2, 2, 4, 0, 2, 6, 0, 6, 5, 0, 4, 4, 2, 4, 0, 4, 0, 0, 8, 2, 4, 8, 0, 4, 2, 2, 6, 0, 0, 6, 0, 4, 2, 0, 2, 0, 2, 8, 0, 8, 0, 0, 8, 0, 5, 4, 0, 8, 2, 0, 12, 0, 2, 10, 0, 4, 2, 0, 4, 0, 0, 10, 0, 6, 2, 0, 2, 0, 0, 4, 0, 6, 0, 0, 14, 0, 2, 2, 0, 2, 2, 0, 2, 2, 2, 4, 0, 8, 4, 0, 10 (list; graph; refs; listen; history; text; internal format)
OFFSET

2,2

COMMENTS

Brent, Hart, Kruppa, and Zimmermann found that a(57885161) = 0. - Charles R Greathouse IV, May 30 2013

LINKS

T. D. Noe, Table of n, a(n) for n = 2..500

Paul Zimmermann, There is no primitive trinomial of degree 57885161 over GF(2), posting to NMBRTHRY mailing list [alternate link]

EXAMPLE

a(7) = 4 because 1 + x + x^7 = 1 + x + x^7, 1 + x^2 + x^7 = (1 + x + x^2)*(1 + x + x^2 + x^4 + x^5), 1 + x^3 + x^7 = 1 + x^3 + x^7, 1 + x^4 + x^7 = 1 + x^4 + x^7, 1 + x^5 + x^7 = (1 + x + x^2)*(1 + x + x^3 + x^4 + x^5) and 1 + x^6 + x^7 = 1 + x^6 + x^7. Thus there are 4 trinomial expressions which cannot be factored over GF(2) and 2 trinomial expressions which do factor.

MATHEMATICA

Do[ c = 0; Do[ If[ ToString[ Factor[ 1 + x^k + x^n, Modulus -> 2 ] ] == ToString[ 1 + x^k + x^n ], c++ ], {k, 1, n - 1} ]; Print[ c ], {n, 2, 100} ]

PROG

(PARI) a(n)=sum(s=1, n-1, polisirreducible((x^n+x^s+1)*Mod(1, 2))) \\ Charles R Greathouse IV, May 30 2013

CROSSREFS

For n such that a(n) > 0 see A073571.

Sequence in context: A131704 A327746 A124492 * A238892 A238279 A282933

Adjacent sequences:  A057643 A057644 A057645 * A057647 A057648 A057649

KEYWORD

nonn,easy,nice

AUTHOR

Robert G. Wilson v, Oct 11 2000

STATUS

approved

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Last modified October 18 12:18 EDT 2019. Contains 328160 sequences. (Running on oeis4.)