With RNA-seq, this can happen due to biases in random hexamer priming during the RT step (explaining the first 6 bases) possibly combined with sequence specificity of the polymerase itself and/or artifacts from end repair (possibly explaining out to 13 bases).
Have you checked whether those first few bases don't belong to any adaptor/barcode sequence ? Normally those sequences if left untrimmed may result into what you have mentioned above. I may be completely wrong but try to go through the FastQC report and if those sequences show up in Over-represented sequences section then you need to trim them off.
The origin of the sample also matters. If the sample preparation isolates certain parts of a genome, for example a CHip-Seq experiment we could expect that to be reflected in the sequence content of the reads.