cp's OEIS Frontend

A376243 Nonnegative integers N = x*y*z = x+y+z for some rational x, y, z.

%I A376243 #26 Sep 27 2024 09:27:56
%S A376243 0,6,7,9,13,14,15,16,19
%N A376243 Nonnegative integers N = x*y*z = x+y+z for some rational x, y, z.
%C A376243 Obviously all of x, y and z must be nonzero for all solutions N > 0. For any N = x*y*z = x+y+z, one gets -N from (-x, -y, -z), so considering only N >= 0 is not a restriction. Either none or exactly two among x, y and z must be negative.
%C A376243 For given N, the problem amounts to finding fractions x and y such that x*y^2 + x*(x - N)*y + N = 0, which in turn corresponds to finding rational points on the elliptic curve Y^2 = X^3 + N^2*(X+4)^2 (with X = -4*N/x and Y = 4*N*D/x^2, where D^2 is the discriminant of the previous quadratic in y).
%C A376243 It appears that (for N > 0) we have a rational solution iff this elliptic curve has nonzero rank. (Is there any counterexample?) If so, the sequence goes (0, 6, 7, 9, 13, 14, 15, 16, 19, 22, 23, 24, 25, 27, 28, 29, 30, 31, 32, 33, 37, 38, 40, 43, 44, 45, 46, 48, 49, 52, 53, 55, 56, 58, 59, 60, 61, 62, 63, 64, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 82, 83, 84, 86, 87, ...)
%H A376243 Allan MacLeod, <a href="https://doi.org/10.48550/arXiv.1610.03430">Elliptic Curves in Recreational Number Theory</a>, arXiv:1610.03430 [math.NT], Oct. 2016.
%H A376243 Victor Miller and others, in reply to Keith F. Lynch, <a href="https://mailman.xmission.com/hyperkitty/list/math-fun@mailman.xmission.com/message/T47KO4SAQEHPTUUX6I6GKZBFU6HS3JSP/">Re: Integer sums and products</a>, math-fun mailing list (available for subscribers), Sep. 2024.
%e A376243 The first few terms correspond to the following solutions (|x| <= |y| <= |z|):
%e A376243     N  |    x    |    y    |    z
%e A376243   -----+---------+---------+---------
%e A376243     0  |    0    |    0    |    0    (or any rational y = -z).
%e A376243     6  |    1    |    2    |    3    (and also {25/21, 54/35, 49/15}).
%e A376243     7  |   7/6   |   4/3   |   9/2
%e A376243     9  |   1/2   |    4    |   9/2
%e A376243    13  |  36/77  | 121/42  | 637/66
%e A376243    14  |   1/3   |    9    |  14/3
%e A376243    15  |   1/2   |   5/2   |   12
%e A376243    16  |  -2/3   |  -4/3   |   18
%e A376243    19  | 121/234 | 324/143 |3211/198
%e A376243   ...
%e A376243 All terms of A054000 (2*n^2-2: 0, 6, 16, 30, 48, 70, 96, 126, 160, 198, ...) are in the sequence, as product and sum of the triple (2*n^2, 1/n - 1, -1/n - 1).
%o A376243 (PARI) select( {is_A376243(n)=!n||ellrank(ellinit([0,1,0,8,16]*n^2))}, [0..30]) \\ Assuming there's a rational solution iff the elliptic curve has rank > 0. - _M. F. Hasler_, Sep 23 2024
%Y A376243 Cf. A376241-A376242 for an enumeration of all possible solutions (not in the order of increasing N) using the Stern-Brocot sequence A002487.
%Y A376243 A054000 (2*n^2-2) is a subsequence.
%K A376243 nonn,more
%O A376243 1,2
%A A376243 _M. F. Hasler_, Sep 16 2024