Predicting next number in a sequence - data analysis
Interesting problem. The pandas autocorrelation plot suggests that the data is random.

How much do you know about the source?  Is it believable that the sequence is, in fact, random? 

Have you gone through the data and plotted a histogram of the integer appearance counts? Do they appear uniformly or are some more frequent than others?

One thing I think you should try is to reformulate your LSTM model. I don't think this is a regression problem, even though the target is a single integer. It's a classification problem with 70 classes. Try treating it that way and using a categorical crossentropy loss function. In such a scenario, ~1.5% accuracy would represent random guessing. Can the model do better than that?