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 2018-08-20, 19:45 #1 henryzz Just call me Henry     "David" Sep 2007 Liverpool (GMT/BST) 32·5·7·19 Posts Concatenated Triangular Numbers Recently someone mentioned to me at a meal that their favourite number was 136 because it was the first 3 triangular numbers concatenated together and was also triangular itself. I wondered whether there were any more numbers with this property. I ran a search today and found no more sets of 3 consecutive triangular numbers that are triangular. Is there any particular reason why this should be the case other than I just haven't searched far enough? I suppose there is pretty good probabilistic argument for there to be no more. The probability of a random number x being triangular is $$O(1/{\sqrt{2x}})$$. The size of the number that needs to be triangular is O(n^6) for the nth set of 3 consecutive triangular numbers concatenated together. I believe that the sum of the probabilities probably converges to a little above 1. Is anyone able to beat my probabilistic argument for there to be no more?

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