A117090 - OEIS

%I #31 Sep 08 2022 08:45:24

%S 10729,5273,2273,829,257,89,73,173,569,1657,4049,8573,16273,28409,

%T 46457,72109,107273,154073,214849,292157,507673,825389,1883773,

%U 2260529,4357673,5834657,8717273,19496657,26342573,31815257,67625969,104356457

%N Primes of the form 9*k^4 - 204*k^3 + 1777*k^2 - 7038*k + 10729, for k >= 0, listed by increasing k.

%C 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.

%D P. Ribenboim, The Book of Prime Number Records. Springer-Verlag, NY, 2nd ed., 1989, p. 137.

%H G. C. Greubel, <a href="/A117090/b117090.txt">Table of n, a(n) for n = 1..5000</a>

%H Carlos Rivera, <a href="http://www.primepuzzles.net/puzzles/puzz_232.htm">Primes and Cubic polynomials</a>, The Prime and Puzzles Connection.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Prime-GeneratingPolynomial.html">Prime-Generating Polynomial</a>.

%F Equals primes of the form ((6*k^2 -68*k +207)^2 + 67)/4, for k>=0. - _G. C. Greubel_, Mar 22 2019

%e For k=0, a(1) = 9*0^4 - 204*0^3 + 1777*0^2 - 7038*0 + 10729 = 10729.

%e For k=1, a(2) = 9*1^4 - 204*1^3 + 1777*1^2 - 7038*1 + 10729 = 5273.

%e For k=2, a(3) = 9*2^4 - 204*2^3 + 1777*2^2 - 7038*2 + 10729 = 2273.

%t Select[Table[9*k^4 - 204*k^3 + 1777*k^2 - 7038*k + 10729, {k, 0, 100}], PrimeQ[#] &] (* _Stefan Steinerberger_, Apr 21 2006 *)

%o (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

%o (PARI) {b(k) = ((6*k^2 -68*k +207)^2+67)/4};

%o for(k=0, 100, if(isprime(b(k)), print1(b(k)", "))) \\ _G. C. Greubel_, Mar 22 2019

%o (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

%Y Cf. A005846, A117624.

%K easy,nonn,less

%O 1,1

%A _Roger L. Bagula_ and Parviz Afereidoon (afereidoon(AT)gmail.com), Apr 18 2006

%E More terms from _Stefan Steinerberger_, Apr 21 2006

%E Edited by _G. C. Greubel_, Mar 22 2019