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%I #67 Sep 19 2022 15:25:14
%S 2,3,5,11,17,41
%N Euler's "Lucky" numbers: n such that m^2-m+n is prime for m=0..n-1.
%C 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 J. H. Conway and R. K. Guy, The Book of Numbers, Copernicus Press, NY, 1996, p. 225.
%D J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 41, p. 16, Ellipses, Paris 2008.
%D I. N. Herstein and I. Kaplansky, Matters Mathematical, Chelsea, NY, 2nd. ed., 1978, see p. 38.
%D F. Le Lionnais, Les Nombres Remarquables. Paris: Hermann, pp. 88 and 144, 1983.
%H 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 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 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/LuckyNumberofEuler.html">Lucky Number of Euler</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Prime-GeneratingPolynomial.html">Prime-Generating Polynomial</a>
%F a(n) = (A003173(n+3) + 1)/4. - _M. F. Hasler_, Nov 03 2008
%t 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 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 (PARI) is(n)=n>1 && qfbclassno(1-4*n)==1 \\ _Charles R Greathouse IV_, Jan 29 2013
%o (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 (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 Cf. A000926, A003173, A092749, A117530, A117531.
%K nonn,fini,full,nice
%O 1,1
%A _Eric W. Weisstein_
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