I urgently need to find out what is happening in regards to below:

I am doing a huge number of simple linear regressions. For each regression I want to use outlier test (outlierTest(fit)) and influence index test and influence plots to identify outliers and influential data points.

I read that for cook's distance people use 1 or 4/n as cutoff. And the outlierTest by default uses 0.05 as cutoff for pvalue.

Doing this, I am getting some data showing that there are no outliers (test result = false with p>0.05) but the cooks distance (using 4/n as cutoff) indicating presence of influential data point.

Can a data point be 'influential' even though it is not an outlier?

Or am I making some mistake in interpreting outlierTest and cook's distance?

Thank you very much!