cp's OEIS Frontend

The polynomial generates 24 primes in absolute value (23 distinct ones) in row starting from n=0 (and 42 primes in absolute value for n from 0 to 46).

The polynomial n^2 - 49*n + 431 generates the same primes in reverse order.

Note: we found in the same family of prime-generating polynomials (with the discriminant equal to 677) the polynomial 13*n^2 - 311*n + 1847 (13*n^2 - 469*n + 4217) generating 23 primes and two noncomposite numbers (in absolute value) in row starting from n=0 (1847, 1549, 1277, 1031, 811, 617, 449, 307, 191, 101, 37, -1, -13, 1, 41, 107, 199, 317, 461, 631, 827, 1049, 1297, 1571, 1871).

Note: another interesting algorithm to produce prime-generating polynomials could be N = m*n^2 + (6*m+1)*n + 8*m + 3, where m, 6*m+1 and 8*m+3 are primes. For m=7 then n=t-20 we get N = 7*t^2 - 237*t + 1999, which generates the following primes: 239, 163, 101, 53, 19, -1, -7, 1, 23, 59, 109, 173, 251 (we can see the same pattern: …, -1, -m, 1, …).

A "prime-generating" polynomial: This polynomial generates 88 distinct primes for 0 <= n <= 99, just two primes fewer than the record held by the polynomial discovered by N. Boston and M. L. Greenwood, that is 41*n^2 - 4641*n + 88007 (this polynomial is sometimes cited as 41*n^2 + 33*n - 43321, which is the same for the input values [-57,42], see the references below).

The nonprime terms in the first 100 are: 10961 = 97*113; 10547 = 53*199; 9353 = 47*199; 7181 = 43*167; 6847 = 41*167; 5893 = 71*83; 3233 = 53*61; 2021 = 43*47; 1681 = 41^2; 1763 = 41*43; 2491 = 47*53; 4331 = 61*71.

For n = m + 41 we obtain the polynomial 4*m^2 - 154*m + 1523, which generates 40 primes in a row starting from m = 0 (polynomial already reported, see the link below).

Inspired by the so-called prime-generating polynomials.

Since p | 3n(n+1) for n=p-1, one has a(n) > p(n). Otherwise stated, a(p-1) = p (as, e.g., for p=23) is optimal.