A328505 Numbers k such that there exist three rationals in arithmetic progression whose product is k.

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

Offset

1,2

Comments

Numbers k such that there exist rationals x, d such that k = x*(x^2-d^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

Robert Israel, Table of n, a(n) for n = 1..1231

Math StackExchange, Prime numbers as the product of an arithmetic sequence

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

Sequence in context: A329500 A287362 A055073 * A012132 A108769 A286754

Adjacent sequences: A328502 A328503 A328504 * A328506 A328507 A328508

Keyword

nonn

Author

Robert Israel, Oct 22 2019

Extensions

More terms from Robert Israel, Mar 02 2020

Status

approved