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A117090
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Primes of the form 9*k^4 - 204*k^3 + 1777*k^2 - 7038*k + 10729, for k >= 0, listed by increasing k.
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2
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10729, 5273, 2273, 829, 257, 89, 73, 173, 569, 1657, 4049, 8573, 16273, 28409, 46457, 72109, 107273, 154073, 214849, 292157, 507673, 825389, 1883773, 2260529, 4357673, 5834657, 8717273, 19496657, 26342573, 31815257, 67625969, 104356457
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OFFSET
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1,1
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COMMENTS
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The Euler polynomial, m^2 + m + q for q=17, generates 16 prime numbers, consecutively, from m=0 to 15. The polynomial 9*k^4 - 204*k^3 + 1777*k^2 - 7038*k + 10729 generates 20 prime numbers, consecutively, for k=0 to 27. The two polynomials are connected by the substitution of m -> 3*k^2 - 34*k + 103 in m^2 + m + 17.
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REFERENCES
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P. Ribenboim, The Book of Prime Number Records. Springer-Verlag, NY, 2nd ed., 1989, p. 137.
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LINKS
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FORMULA
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Equals primes of the form ((6*k^2 -68*k +207)^2 + 67)/4, for k>=0. - G. C. Greubel, Mar 22 2019
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EXAMPLE
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For k=0, a(1) = 9*0^4 - 204*0^3 + 1777*0^2 - 7038*0 + 10729 = 10729.
For k=1, a(2) = 9*1^4 - 204*1^3 + 1777*1^2 - 7038*1 + 10729 = 5273.
For k=2, a(3) = 9*2^4 - 204*2^3 + 1777*2^2 - 7038*2 + 10729 = 2273.
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MATHEMATICA
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Select[Table[9*k^4 - 204*k^3 + 1777*k^2 - 7038*k + 10729, {k, 0, 100}], PrimeQ[#] &] (* Stefan Steinerberger, Apr 21 2006 *)
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PROG
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(Magma) [a: k in [0..100] | IsPrime(a) where a is 9*k^4 - 204*k^3 + 1777*k^2 - 7038*k + 10729]; // Vincenzo Librandi, Sep 17 2015
(PARI) {b(k) = ((6*k^2 -68*k +207)^2+67)/4};
for(k=0, 100, if(isprime(b(k)), print1(b(k)", "))) \\ G. C. Greubel, Mar 22 2019
(Sage) b(k)=((6*k^2 -68*k +207)^2+67)/4; [b(k) for k in (0..100) if is_prime(b(k))] # G. C. Greubel, Mar 22 2019
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CROSSREFS
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KEYWORD
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easy,nonn,less
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AUTHOR
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Roger L. Bagula and Parviz Afereidoon (afereidoon(AT)gmail.com), Apr 18 2006
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EXTENSIONS
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STATUS
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approved
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