Time series data: How I measure influence of new product sales on existing product sales (statistically)?
You could build an ARIMAX model. This would permit to include autoregressive (AR) terms as well as well as the sales in product 5 as an Exogenous Input (X). This would give you a potential model where the sales for a product $i$ at time $t$ is given by $s^i_t$ then,

$s^1_t=\alpha_1 s^1_{t-1} + \alpha_2 s^1_{t-2} + \ldots + \beta_0 s^5_t + \beta_1 s^5_{t-1} + \ldots $

Note that you may need to make the series stationary first, but see more on that below. You could estimate this model with the seasonal R package that relies on the X-13ARIMA-SEATS software developed by the US Census Bureau.

I would recommend to ensure that your time series are all stationary, see for example this post before you use X13. I would also run cointegration tests. See for more explanation this excellent post.

Since you only have data on week 5 I would start modeling in week 5 but you could include autoregressive (AR) terms related to the sales of product 1 prior to week 5.