A225768 Least k > 0 such that k^6 + n is prime, or 0 if k^6 + n is never prime.

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

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 * A335024 A270927 A089512

Adjacent sequences: A225765 A225766 A225767 * A225769 A225770 A225771

Keyword

nonn

Author

M. F. Hasler, Jul 25 2013

Status

approved