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 A231613 Numbers n such that the four sixth-degree cyclotomic polynomials are simultaneously prime. 4
 32034, 162006, 339105, 458811, 1780425, 2989119, 2993100, 3080205, 4375404, 6129597, 6280221, 7565142, 8489820, 10268277, 11343741, 12065076, 13067295, 13333182, 15866508, 16472802, 17040537, 18028605, 19066758, 22633629, 24256362, 24365259, 25031349 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS The polynomials are cyclotomic(7,x) = 1 + x + x^2 + x^3 + x^4 + x^5 + x^6, cyclotomic(9,x) = 1 + x^3 + x^6, cyclotomic(14,x) = 1 - x + x^2 - x^3 + x^4 - x^5 + x^6, and cyclotomic(18,x) = 1 - x^3 + x^6. The numbers 7, 9, 14 and 18 are in the sixth row of A032447. By Schinzel's hypothesis H, there are an infinite number of n that yield simultaneous primes. Note that the two first-degree cyclotomic polynomials, x-1 and x+1, yield the twin primes for the numbers in A014574. REFERENCES See A087277. LINKS Table of n, a(n) for n=1..27. MATHEMATICA t = {}; n = 0; While[Length[t] < 30, n++; If[PrimeQ[Cyclotomic[7, n]] && PrimeQ[Cyclotomic[9, n]] && PrimeQ[Cyclotomic[14, n]] && PrimeQ[Cyclotomic[18, n]], AppendTo[t, n]]]; t Select[Range[251*10^5], AllTrue[Cyclotomic[{7, 9, 14, 18}, #], PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Oct 29 2016 *) CROSSREFS Cf. A014574 (first degree solutions: average of twin primes). Cf. A087277 (similar, but with second-degree cyclotomic polynomials). Cf. A231612 (similar, but with fourth-degree cyclotomic polynomials). Cf. A231614 (similar, but with eighth-degree cyclotomic polynomials). Sequence in context: A269319 A197114 A224621 * A054038 A156977 A217368 Adjacent sequences: A231610 A231611 A231612 * A231614 A231615 A231616 KEYWORD nonn AUTHOR T. D. Noe, Dec 11 2013 STATUS approved

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Last modified May 21 17:21 EDT 2024. Contains 372738 sequences. (Running on oeis4.)