A120436 Complete list of numbers that can be represented both as a product of 2 consecutive integers and as a product of 3 consecutive integers.

0, 6, 210

Offset

1,2

Comments

Mordell shows that the only integer points on the elliptic curve y*(y+1) = x*(x+1)*(x+2) have x = -2, -1, 0, 1, 5, corresponding to the products 0, 6, 210. However, there are infinitely many rational points generated from (x,y) = (0,0) by the chord-and-tangent process. - Jonathan Sondow, Oct 12 2013

References

Louis J. Mordell, Diophantine Equations, Academic Press 1969, p. 257.

Links

Table of n, a(n) for n=1..3.

Formula

Terms can be derived from the integral solutions to the elliptic curve y^2 = x^3 - 16*x + 16.

Example

210 = 14*15 = 5*6*7.

Mathematica

Module[{nn=20, p2, p3}, p2=Times@@@Partition[Range[0, nn], 2, 1]; p3= Times@@@ Partition[ Range[0, nn], 3, 1]; Intersection[p2, p3]] (* Harvey P. Dale, Oct 05 2019 *)

Crossrefs

Intersection of A002378 and A007531.

Sequence in context: A024083 A172530 A201237 * A136742 A334285 A068969

Adjacent sequences: A120433 A120434 A120435 * A120437 A120438 A120439

Keyword

fini,full,nonn,bref

Author

Andrew Niedermaier, Jul 16 2006

Status

approved