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A225768 Least k > 0 such that k^6 + n is prime, or 0 if k^6 + n is never prime. 6
0, 1, 1, 2, 1, 18, 1, 2, 0, 2, 1, 54, 1, 28, 3, 2, 1, 18, 1, 2, 399, 26, 1, 6, 5, 2, 21, 0, 1, 288, 1, 4, 3, 2, 105, 6, 1, 2, 33, 2, 1, 546, 1, 2, 3, 2, 1, 6, 35, 2, 51, 20, 1, 12, 5, 28, 9, 4, 1, 18, 1, 4, 63, 2, 0, 18, 1, 2, 3, 28, 1, 6, 1, 2, 15, 2, 35, 24, 1, 12, 3, 4, 1, 42, 115, 2, 111, 2, 1, 18, 91, 6, 3, 2, 3, 6, 1, 28, 3, 2 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

Motivated by the "particularly poor polynomial" n^6+1091 (composite for n=1,...,3905) mentioned on Weisstein's page about prime generating polynomials.

We have a(n) = 0 if n is a cube n = g^3 with g > 1 because then k^6 + g^3 = (k^2 + g)*(k^4 - k^2*g + g^2) ), which can be prime only when n = g = 1. - T. D. Noe, Nov 18 2013

By the theorem of Brillhart, Filaseta and Odlyzko (see link), if a(n) > n > 1 then x^6 + n must be irreducible.  If x^6 + n is irreducible, the Bunyakovsky conjecture implies a(n) is finite. - Robert Israel, Apr 25 2016

LINKS

Robert Israel, Table of n, a(n) for n = 0..10000

J. Brillhart, M. Filaseta, A. Odlyzko, On an irreducibility theorem of A. Cohn. Canadian Journal of Mathematics 33(1981), 1055-1059.

Eric Weisstein's World of Mathematics, Prime-Generating Polynomial.

MAPLE

f:= proc(n) local exact, x, k, F, nf, F1, C;

    iroot(n, 3, exact);

    if exact and n > 1 then return 0 fi;

    if irreduc(x^6+n) then

       for k from 1+(n mod 2) by 2 do if isprime(k^6+n) then return k fi od

    else

       F:= factors(x^6+n)[2]; #

       F1:= map(t -> t[1], F);

       nf:= nops(F);

       C:= map(t -> op(map(rhs@op, {isolve(t^2-1)})), F1);

       for k in sort(convert(select(type, C, positive), list)) do

         if isprime(k^6+n) then return k fi

       od:

       0

    fi

end proc:

map(f, [$0..100]); # Robert Israel, Apr 25 2016

MATHEMATICA

{0, 1}~Join~Table[If[IrreduciblePolynomialQ[x^6 + n], SelectFirst[Range[1 + Mod[n, 2], 10^3, 2], PrimeQ[#^6 + n] &], 0], {n, 2, 120}] (* Michael De Vlieger, Apr 25 2016, Version 10 *)

PROG

(PARI) {(a, b=6)->#factor(x^b+a)~==1 & for(n=1, 9e9, ispseudoprime(n^b+a)&return(n)); a==1&return(1); print1("/*"a":", factor(x^b+a)"*/")} /* For illustrative purpose only. The polynomial x^6+a is factored to avoid an infinite loop when it is composite. But there could be x such that this is prime, when all factors but one are 1 (not for exponent b=6, but, e.g., x=4 for exponent b=4), see A225766. */

CROSSREFS

See A085099, A225765 - A225770 for the k^2, k^3, ..., k^8 analogs.

Sequence in context: A013072 A012895 A013077 * A270927 A089512 A300956

Adjacent sequences:  A225765 A225766 A225767 * A225769 A225770 A225771

KEYWORD

nonn

AUTHOR

M. F. Hasler, Jul 25 2013

STATUS

approved

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Last modified January 17 18:08 EST 2020. Contains 330987 sequences. (Running on oeis4.)