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 A014556 #80 Feb 16 2025 08:32:33
%S A014556 2,3,5,11,17,41
%N A014556 Euler's "Lucky" numbers: n such that m^2-m+n is prime for m=0..n-1.
%C A014556 Same as n such that 4n-1 is a Heegner number 1,2,3,7,11,19,43,67,163 (see A003173 and Conway and Guy's book).
%D A014556 J. H. Conway and R. K. Guy, The Book of Numbers, Copernicus Press, NY, 1996, p. 225.
%D A014556 J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 41, p. 16, Ellipses, Paris 2008.
%D A014556 I. N. Herstein and I. Kaplansky, Matters Mathematical, Chelsea, NY, 2nd. ed., 1978, see p. 38.
%D A014556 F. Le Lionnais, Les Nombres Remarquables. Paris: Hermann, pp. 88 and 144, 1983.
%H A014556 Aram Bingham, <a href="https://arxiv.org/abs/2002.02059">Ternary arithmetic, factorization, and the class number one problem</a>, arXiv:2002.02059 [math.NT], 2020. See p. 8.
%H A014556 Hung Viet Chu, Steven J. Miller, and Joshua M. Siktar, <a href="https://arxiv.org/abs/2411.03330">Composite Numbers in an Arithmetic Progression</a>, arXiv:2411.03330 [math.HO], 2024. See p. 7.
%H A014556 Brady Haran and Matt Parker, <a href="https://www.youtube.com/watch?v=gM5uNcgn2NQ">Caboose Numbers</a>, Youtube video, June 2024.
%H A014556 Harold M. Stark, <a href="https://doi.org/10.1307/mmj/1028999653">A complete determination of the complex quadratic fields of class-number one</a>, The Michigan Mathematical Journal 14.1 (1967): 1-27.
%H A014556 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LuckyNumberofEuler.html">Lucky Number of Euler</a>
%H A014556 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Prime-GeneratingPolynomial.html">Prime-Generating Polynomial</a>
%F A014556 a(n) = (A003173(n+3) + 1)/4. - _M. F. Hasler_, Nov 03 2008
%t A014556 A003173 = Union[Select[-NumberFieldDiscriminant[Sqrt[-#]] & /@ Range[200], NumberFieldClassNumber[Sqrt[-#]] == 1 &] /. {4 -> 1, 8 -> 2}]; a[n_] := (A003173[[n + 4]] + 1)/4; Table[a[n], {n, 0, 5}] (* _Jean-François Alcover_, Jul 16 2012, after _M. F. Hasler_ *)
%t A014556 Select[Range[50],AllTrue[Table[m^2-m+#,{m,0,#-1}],PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* _Harvey P. Dale_, May 12 2017 *)
%o A014556 (PARI) is(n)=n>1 && qfbclassno(1-4*n)==1 \\ _Charles R Greathouse IV_, Jan 29 2013
%o A014556 (PARI) is(p)=for(n=1,p-1, if(!isprime(n*(n-1)+p),return(0))); 1 \\ naive; _Charles R Greathouse IV_, Aug 26 2022
%o A014556 (PARI) is(p)=for(n=1,sqrt(p/3)\/1, if(!isprime(n*(n-1)+p),return(0))); 1 \\ _Charles R Greathouse IV_, Aug 26 2022
%Y A014556 Cf. A000926, A003173, A092749, A117530, A117531.
%K A014556 nonn,fini,full,nice
%O A014556 1,1
%A A014556 _Eric W. Weisstein_