A014556 Euler's "Lucky" numbers: n such that m^2-m+n is prime for m=0..n-1.

2, 3, 5, 11, 17, 41

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

Table of n, a(n) for n=1..6.

Aram Bingham, Ternary arithmetic, factorization, and the class number one problem, arXiv:2002.02059 [math.NT], 2020. See p. 8.

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

Cf. A000926, A003173, A092749, A117530, A117531.

Sequence in context: A079370 A014210 A203074 * A062737 A085613 A082605

Adjacent sequences: A014553 A014554 A014555 * A014557 A014558 A014559

Keyword

nonn,fini,full,nice

Author

Eric W. Weisstein

Status

approved