A120436 - OEIS

%I #14 Oct 05 2019 11:53:02

%S 0,6,210

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

%C 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

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

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

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

%t 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 *)

%Y Intersection of A002378 and A007531.

%K fini,full,nonn,bref

%O 1,2

%A _Andrew Niedermaier_, Jul 16 2006