Slow Weekend......?

Please give at least the next 5 terms of this sequence...........

4, 7, 8, 11, 16, 19, 24, 27, 38, 41.........etc.

I asume:

$\begin{array}{|l|r|r|r|r|r|r|r|r|r|r|r|r|r|r|r|} \hline &4&{\color{green}7}&8&11&16&19&24&2{\color{green}7}&38&41&{\color{grey}48}&{\color{grey}53}&{\color{grey}56}&{\color{grey}61}&{\color{grey}68} \\ \hline \text{Prime numbers}&2&3&{\color{blue}5}&{\color{blue}7}&{\color{blue}11}&{\color{blue}13}&{\color{blue}17}&{\color{blue}19}&23&29&31&37&41&43&47 \\ \text{Prime numbers}&{\color{red}3}&{\color{red}5}&7&11&13&17&19&23&{\color{red}29}&{\color{red}31}&{\color{red}37}&{\color{red}41}&{\color{red}43}&{\color{red}47}&{\color{red}53}\\ \hline \text{consecutive numbering}&{\color{red}1}&{\color{red}2}&{\color{blue}3}&{\color{blue}4}&{\color{blue}5}&{\color{blue}6}&{\color{blue}7}&{\color{blue}8}&{\color{red}9}&{\color{red}10}&{\color{red}11}&{\color{red}12}&{\color{red}13}&{\color{red}14}&{\color{red}15}\\ \hline \text{add red or add blue }& \\ \text{change with digit seven} &\\ \hline \end{array}$

Thank you heureka: The sequence is very simple; just subtract the counting numbers from the sequence and you you get? Twin Primes!!. Or natural counting numbers added to Twin Primes:

1, 2, 3, 4, 5, 6.......+ 3, 5.....5, 7.......11,13..................etc.