This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A212325 #39 Feb 16 2025 08:33:17
%S A212325 -167,-163,-157,-149,-139,-127,-113,-97,-79,-59,-37,-13,13,41,71,103,
%T A212325 137,173,211,251,293,337,383,431,481,533,587,643,701,761,823,887,953,
%U A212325 1021,1091,1163,1237,1313,1391,1471,1553,1637,1723,1811,1901,1993,2087,2183,2281,2381
%N A212325 Prime-generating polynomial: a(n) = n^2 + 3*n - 167.
%C A212325 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).
%C A212325 The polynomial n^2 - 49*n + 431 generates the same primes in reverse order.
%C A212325 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).
%C A212325 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, …).
%H A212325 Bruno Berselli, <a href="/A212325/b212325.txt">Table of n, a(n) for n = 0..1000</a>
%H A212325 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Prime-GeneratingPolynomial.html">Prime-Generating Polynomial</a>.
%H A212325 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F A212325 G.f.: (-167 + 338*x - 169x^2)/(1-x)^3. - _Bruno Berselli_, May 18 2012
%F A212325 From _Elmo R. Oliveira_, Feb 10 2025: (Start)
%F A212325 E.g.f.: exp(x)*(-167 + 4*x + x^2).
%F A212325 a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2. (End)
%t A212325 Table[n^2+3n-167,{n,0,50}] (* or *) LinearRecurrence[{3,-3,1},{-167,-163,-157},50] (* _Harvey P. Dale_, Feb 08 2020 *)
%o A212325 (Magma) [n^2+3*n-167: n in [0..47]]; // _Bruno Berselli_, May 18 2012
%o A212325 (PARI) Vec((-167+338*x-169*x^2)/(1-x)^3+O(x^99)) \\ _Charles R Greathouse IV_, Oct 01 2012
%o A212325 (PARI) apply( {A212325(n)=(n+3)*n-167}, [0..55]) \\ _M. F. Hasler_, Feb 11 2025
%o A212325 (Python) def A212325(n=None, upto=None): return(A212325(i)for i in range(n or 0, upto or 2**63)) if upto or n is None else(n+3)*n-167 # _M. F. Hasler_, Feb 11 2025
%Y A212325 Cf. A060566 (an 80 primes generating pol.), A202018 (Euler's p.g.p.), A050268, A181963, A181973, A182409, A211773, A318791, A320772, A330363 (other p.g.p.).
%K A212325 sign,easy
%O A212325 0,1
%A A212325 _Marius Coman_, May 14 2012
%E A212325 Edited by _Bruno Berselli_, May 18 2012