

A328505


Numbers k such that there exist three rationals in arithmetic progression whose product is k.


1



1, 3, 6, 8, 10, 11, 13, 14, 15, 17, 21, 24, 25, 27, 28, 29, 31, 34, 35, 36, 39, 42, 43, 45, 46, 47, 48, 49, 52, 53, 55, 57, 60, 62, 63, 64, 65, 66, 67, 68, 70, 71, 76, 78, 80, 81, 82, 83, 85, 86, 88, 89, 90, 91, 93, 99, 101, 103, 104, 105, 106, 107, 109, 111, 112, 114, 118, 119, 120, 122, 123
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OFFSET

1,2


COMMENTS

Numbers k such that there exist rationals x, d such that k = x*(x^2d^2).
Numbers k such that the elliptic curve s^3 + t^2  k^2 has rational points other than (s,t) = (0, +/ k). Namely, if (s,t) is a rational point on the curve with s <> 0, then we can take x=k/s, d=t/s.
If k is in the sequence, then so is j^3*k for any j.


LINKS



EXAMPLE

a(5) = 10 is in the sequence because (1, 5/2, 4) are three rationals in arithmetic progression whose product is 10.
a(6) = 11 is in the sequence because (9/2,11/12,8/3) are three rationals in arithmetic progression whose product is 11.


PROG

(Sage)
R = range(1, 100);
S = [];
for r in R:
E = EllipticCurve([0, 0, 0, 0, r^2]);
if (E.torsion_subgroup().order() > 3) or (E.rank(only_use_mwrank=False) > 0):
S.append(r);
S


CROSSREFS



KEYWORD

nonn


AUTHOR



EXTENSIONS



STATUS

approved



