A003585 - OEIS

%I #41 Jul 12 2024 09:51:41

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

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

%C Under GRH the terms shown are the complete sequence.

%C 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

%C Duke & Schulze-Pillot showed that this sequence is finite. - _Charles R Greathouse IV_, Jul 12 2024

%D 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).

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

%D 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.

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

%H Amir Jafari and Farhood Rostamkhani, <a href="https://arxiv.org/abs/2109.10225">On ternary quadratic forms over the rational numbers</a>, arXiv:2109.10225 [math.HO], 2021. See p. 3.

%H Ben Kane, <a href="https://arxiv.org/abs/0903.3026">Representing Sets With Sums Of Triangular Numbers</a>, Int. Math. Res. Not., Vol. 2009, No. 17, pp. 3264-3285, arXiv:0903.3026 [math.NT], 2009.

%H K. Ono, <a href="http://www.jstor.org/stable/2974471">Ramanujan, taxicabs, birthdates, ZIP codes and twists</a>, Amer. Math. Monthly, 104 (1997), 912-917.

%H K. Ono, <a href="http://www.ams.org/notices/200606/fea-ono.pdf">Honoring a gift from Kumbakonam</a>, Notices Amer. Math. Soc., 53 (2006), 640-651.

%t 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 *)

%Y Cf. A002145.

%K nonn,fini,full,nice

%O 1,1

%A _N. J. A. Sloane_