A335715 - OEIS

%I #28 Jul 29 2020 03:25:54

%S 6,210,546,915,1785,7230,13395,16206,17490,20930,76245,104006,228486,

%T 508530,563766

%N Squarefree numbers m such that the equation x*(x+1)*(x+2) = m*y^2 has more than one solution (x,y) with x>0 and y>0.

%C 7453355, 9147666, 19073886, 61622814, 648402306, 22023958326, 748181603526 are also terms. - _Jinyuan Wang_, Jul 28 2020

%H Michael A. Bennett, <a href="http://dx.doi.org/10.4064/aa105-4-3">Lucas' square pyramid problem revisited</a>, Acta Arithmetica 105 (2002), 341-347.

%H Michel Marcus and Jinyuan Wang, <a href="/A335715/a335715.txt">PARI program</a>

%e 6 is a term since x*(x+1)*(x+2) = 6*y^2 has 3 positive integer solutions : (1,1), (2,2) and (48,140) that give (1,1) and (24,70) for s*(s+1)*(2s+1) = 6*t^2 with a (x=2s, y=2t) variable change (see A000330 where A000330(n) = n*(n+1)*(2*n+1)/6).

%Y Cf. A000330 (square pyramidal numbers), A005117 (squarefree numbers).

%Y Cf. A335785 (at least one solution), A336145.

%K nonn,more

%O 1,1

%A _Michel Marcus_, Jun 18 2020

%E a(12) from _Michel Marcus_, Jun 24 2020

%E a(13)-a(15) from _Jinyuan Wang_, Jul 28 2020