Valid or not valid: that is the question

$\large {\begin{eqnarray} 3+4&=&7 \\ 11+12&=&23 \end{eqnarray} }$

A certain sequence of consecutive numbers, with first term as a prime number, can have its sum equal to a prime number. The examples above demonstrate this property with two different sequences, each with two consecutive numbers, and the first ones are prime numbers. Is this property valid for sequences like these with (strictly) more than two terms?

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