A003585 Odd numbers that are not of the form x^2 + y^2 + 10*z^2.

3, 7, 21, 31, 33, 43, 67, 79, 87, 133, 217, 219, 223, 253, 307, 391, 679, 2719

Offset

1,1

Comments

Under GRH the terms shown are the complete sequence.

The prime numbers in this sequence (3,7,31,43,67,79,223,307,2719) are of the form 4n+3. - Paul Muljadi, Jan 28 2011

Duke & Schulze-Pillot showed that this sequence is finite. - Charles R Greathouse IV, Jul 12 2024

References

William Duke and Rainer Schulze-Pillot, Representation of integers by positive ternary quadratic forms and equidistribution of lattice points on ellipsoids, Inventiones mathematicae, Volume 99, pages 49-57, (1990).

K. Ono, Ramanujan's ternary quadratic form, Abstract 914-11-96, Abstracts Amer. Math. Soc., 17 (1996), 476.

Ken Ono and Amir D. Aczel, "My search for Ramanujan", Springer, 2016; see p. 182, lines 12 and 11 from the bottom. but there is an error.

S. Ramanujan, Coll. Papers, p. 172.

Links

Table of n, a(n) for n=1..18.

Amir Jafari and Farhood Rostamkhani, On ternary quadratic forms over the rational numbers, arXiv:2109.10225 [math.HO], 2021. See p. 3.

Ben Kane, Representing Sets With Sums Of Triangular Numbers, Int. Math. Res. Not., Vol. 2009, No. 17, pp. 3264-3285, arXiv:0903.3026 [math.NT], 2009.

K. Ono, Ramanujan, taxicabs, birthdates, ZIP codes and twists, Amer. Math. Monthly, 104 (1997), 912-917.

K. Ono, Honoring a gift from Kumbakonam, Notices Amer. Math. Soc., 53 (2006), 640-651.

Mathematica

r[n_, z_] := Reduce[n == x^2 + y^2 + 10*z^2, {x, y}, Integers]; A003585 = Reap[For[n=1, n<3000, n = n+2, If[Union[Table[r[n, z], {z, 0, Sqrt[n/10] // Ceiling}]] == {False}, Print[n]; Sow[n]]]][[2, 1]] (* Jean-François Alcover, Jul 06 2011, updated Jan 27 2015 *)

Crossrefs

Cf. A002145.

Sequence in context: A090504 A018548 A323585 * A108102 A357898 A065523

Adjacent sequences: A003582 A003583 A003584 * A003586 A003587 A003588

Keyword

nonn,fini,full,nice

Author

N. J. A. Sloane

Status

approved