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A268101 Smallest prime p such that some polynomial of the form a*x^2 - b*x + p generates distinct primes in absolute value for x = 1 to n, where 0 < a < p and 0 <= b < p.

%I A268101 #19 Feb 16 2025 08:33:30
%S A268101 2,3,5,5,7,7,11,11,11,11,13,13,17,17,17,17,19,19,23,23,23,23,29,29,29,
%T A268101 29,29,29,31,41,41,41,41,41,41,41,41,41,41,41,647,1277,1979,2753
%N A268101 Smallest prime p such that some polynomial of the form a*x^2 - b*x + p generates distinct primes in absolute value for x = 1 to n, where 0 < a < p and 0 <= b < p.
%H A268101 Carlos Rivera, <a href="http://www.primepuzzles.net/problems/prob_012.htm">Problem 12</a>
%H A268101 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Prime-GeneratingPolynomial.html">Prime-Generating Polynomial</a>
%e A268101 a(1) = 2 (a prime), x^2 + 2 gives a prime for x = 1.
%e A268101 a(2) = 3 (a prime), 2*x^2 + 3 gives distinct primes for x = 1 to 2.
%e A268101 a(4) = 5 (a prime), 2*x^2 + 5 gives distinct primes for x = 1 to 4.
%e A268101 a(6) = 7 (a prime), 4*x^2 + 7 gives distinct primes for x = 1 to 6.
%e A268101 a(10) = 11 (a prime), 2*x^2 + 11 gives distinct primes for x = 1 to 10.
%e A268101 a(12) = 13 (a prime), 6*x^2 + 13 gives distinct primes for x = 1 to 12.
%e A268101 a(16) = 17 (a prime), 6*x^2 + 17 gives distinct primes for x = 1 to 16.
%e A268101 a(18) = 19 (a prime), 10*x^2 + 19 gives distinct primes for x = 1 to 18.
%e A268101 a(22) = 23 (a prime), 3*x^2 - 3*x + 23 gives distinct primes for x = 1 to 22.
%e A268101 a(28) = 29 (a prime), 2*x^2 + 29 gives distinct primes for x = 1 to 28.
%e A268101 a(29) = 31 (a prime), 2*x^2 - 4*x + 31 gives distinct primes for x = 1 to 29.
%e A268101 a(40) = 41 (a prime), x^2 - x + 41 gives distinct primes for x = 1 to 40.
%e A268101 a(41) = 647 (a prime), abs(36*x^2 - 594*x + 647) gives distinct primes for x = 1 to 41.
%e A268101 a(42) = 1277 (a prime), abs(36*x^2 - 666*x + 1277) gives distinct primes for x = 1 to 42.
%e A268101 a(43) = 1979 (a prime), abs(36*x^2 - 738*x + 1979) gives distinct primes for x = 1 to 43.
%e A268101 a(44) = 2753 (a prime), abs(36*x^2 - 810*x + 2753) gives distinct primes for x = 1 to 44.
%Y A268101 Cf. A027688, A027753, A027690, A027755, A048058, A048059, A007635, A007639, A007637, A007641, A202018, A005846, A117081, A050268, A268109.
%K A268101 nonn,hard
%O A268101 1,1
%A A268101 _Arkadiusz Wesolowski_, Jan 26 2016