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A225768
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Least k > 0 such that k^6 + n is prime, or 0 if k^6 + n is never prime.
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6
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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
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OFFSET
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0,4
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COMMENTS
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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
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LINKS
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MAPLE
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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:
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MATHEMATICA
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{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 *)
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PROG
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(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. */
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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