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%I #25 Mar 02 2020 18:52:37
%S 1,3,6,8,10,11,13,14,15,17,21,24,25,27,28,29,31,34,35,36,39,42,43,45,
%T 46,47,48,49,52,53,55,57,60,62,63,64,65,66,67,68,70,71,76,78,80,81,82,
%U 83,85,86,88,89,90,91,93,99,101,103,104,105,106,107,109,111,112,114,118,119,120,122,123
%N Numbers k such that there exist three rationals in arithmetic progression whose product is k.
%C Numbers k such that there exist rationals x, d such that k = x*(x^2-d^2).
%C 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.
%C If k is in the sequence, then so is j^3*k for any j.
%H Robert Israel, <a href="/A328505/b328505.txt">Table of n, a(n) for n = 1..1231</a>
%H Math StackExchange, <a href="https://math.stackexchange.com/questions/3401490/">Prime numbers as the product of an arithmetic sequence</a>
%e a(5) = 10 is in the sequence because (1, 5/2, 4) are three rationals in arithmetic progression whose product is 10.
%e a(6) = 11 is in the sequence because (-9/2,-11/12,8/3) are three rationals in arithmetic progression whose product is 11.
%o (Sage)
%o R = range(1,100);
%o S = [];
%o for r in R:
%o E = EllipticCurve([0,0,0,0,r^2]);
%o if (E.torsion_subgroup().order() > 3) or (E.rank(only_use_mwrank=False) > 0):
%o S.append(r);
%o S
%K nonn
%O 1,2
%A _Robert Israel_, Oct 22 2019
%E More terms from _Robert Israel_, Mar 02 2020
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