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\hline  p & \mathrm{OEIS\;ID\;for\;}p\mathrm{-rough\;numbers.} \\  \hline  2 & A000027\\ A000027 & 1,2,3,4,5,6,7\\  3 & A005408\\ A005408 & 1,3,5,7,9,11,13\\  5 & A007310\\ A007310 & 1,5,7,11,13,17,19\\  7 & A007775\\ A007775 & 1, 7, 11, 13, 17, 19, 23\\  11 & A008364\\ A008364& 1, 11, 13, 17, 19, 23, 29\\  13 & A008365\\ A008365& 1, 13, 17, 19, 23, 29, 31\\  17 & A008366\\ A008366& 1, 17, 19, 23, 29, 31, 37,\\  19 & A166061\\ A166061& 1, 19, 23, 29, 31, 37, 41\\  23 & A166063\\ A166063& 1, 23, 29, 31, 37, 41, 43\\  \hline  \end{array}  \end{equation}  From the table above we can see the first entry is always $1$, the second entry is the $n^{th}$ prime, (i.e the index in the name of that sequence) and the next entry is the name of the next sequence. There are an increasing number of primes in each sequence, rather the probability of finding a prime number is higher.  \section{Generating Functions}  A generating function for a sequence with terms $a(n)$ is simply the series, \begin{equation}  G_a(z)=a(0)z^0 + a(1)z^1 + a(2)z^2 + \cdots= \sum_{n=0}^\infty a(n)z^n,