The number 185136 has the interesting property that it is a triangular number that is also the product of 3 consecutive integers.
I square 185136 and each of the x consecutive numbers following it, and total the sum of all these squares, S.
Then I do the same with 185136+x+1 and the y consecutive numbers following it, until I again reach the same sum, S.
Find positive integer values of x,y,S compliant with the above requirements.
(In reply to re: Bit different with added consideration
Which triangle number is 185136?
I find n=((√8*T) - 1)/2.
Substituting for T gives n=608 which is twice the value required for the x value of 304.
Again I find that the sum of the first n squares is given by:
n³/3 + n²/2 + n/6.
On that logic it seems that I need the sum of the first (n+304) squares minus the sum of the first n  squares to arrive at: 10471153462280.
I'm assuming that is correct without going "through the hoops".
Having shown in a prior comment that y is x-1 one would do similarly for the ensuing the sum of the next x-1 squares arrives at the same result.
I am extrapolating upon previous findings and internet findings but I feel this is right.
Posted by brianjn
on 2013-01-08 06:48:29