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A117090
Primes of the form 9*k^4 - 204*k^3 + 1777*k^2 - 7038*k + 10729, for k >= 0, listed by increasing k.
2
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
OFFSET
1,1
COMMENTS
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.
REFERENCES
P. Ribenboim, The Book of Prime Number Records. Springer-Verlag, NY, 2nd ed., 1989, p. 137.
LINKS
Carlos Rivera, Primes and Cubic polynomials, The Prime and Puzzles Connection.
Eric Weisstein's World of Mathematics, Prime-Generating Polynomial.
FORMULA
Equals primes of the form ((6*k^2 -68*k +207)^2 + 67)/4, for k>=0. - G. C. Greubel, Mar 22 2019
EXAMPLE
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.
MATHEMATICA
Select[Table[9*k^4 - 204*k^3 + 1777*k^2 - 7038*k + 10729, {k, 0, 100}], PrimeQ[#] &] (* Stefan Steinerberger, Apr 21 2006 *)
PROG
(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
CROSSREFS
Sequence in context: A238067 A184603 A247854 * A255038 A255031 A256949
KEYWORD
easy,nonn,less
AUTHOR
Roger L. Bagula and Parviz Afereidoon (afereidoon(AT)gmail.com), Apr 18 2006
EXTENSIONS
More terms from Stefan Steinerberger, Apr 21 2006
Edited by G. C. Greubel, Mar 22 2019
STATUS
approved