OFFSET
1,1
COMMENTS
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).
REFERENCES
J. H. Conway and R. K. Guy, The Book of Numbers, Copernicus Press, NY, 1996, p. 225.
J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 41, p. 16, Ellipses, Paris 2008.
I. N. Herstein and I. Kaplansky, Matters Mathematical, Chelsea, NY, 2nd. ed., 1978, see p. 38.
F. Le Lionnais, Les Nombres Remarquables. Paris: Hermann, pp. 88 and 144, 1983.
LINKS
Aram Bingham, Ternary arithmetic, factorization, and the class number one problem, arXiv:2002.02059 [math.NT], 2020. See p. 8.
Hung Viet Chu, Steven J. Miller, and Joshua M. Siktar, Composite Numbers in an Arithmetic Progression, arXiv:2411.03330 [math.HO], 2024. See p. 7.
Brady Haran and Matt Parker, Caboose Numbers, Youtube video, June 2024.
Harold M. Stark, A complete determination of the complex quadratic fields of class-number one, The Michigan Mathematical Journal 14.1 (1967): 1-27.
Eric Weisstein's World of Mathematics, Lucky Number of Euler
Eric Weisstein's World of Mathematics, Prime-Generating Polynomial
FORMULA
a(n) = (A003173(n+3) + 1)/4. - M. F. Hasler, Nov 03 2008
MATHEMATICA
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 *)
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 *)
PROG
(PARI) is(n)=n>1 && qfbclassno(1-4*n)==1 \\ Charles R Greathouse IV, Jan 29 2013
(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
(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
CROSSREFS
KEYWORD
nonn,fini,full,nice
AUTHOR
STATUS
approved