Please give the next 4 terms of this sequence. Thanks for any help:
2, 4, 7, 10, 15, 18, 23, 26............640 (is 100th term)
Thanks heureka: brilliant work!. In my book it is stated slightly different: It is the sequence of natual or counting numbers added to prime numbers - 1. So, we have:
Counting numbers: 1, 2, 3, 4, 5, 6............
Prime numbers + 2, 3, 5, 7, 11, 13...............
Subtract 1 from 2, 4, 7, 10, 15, 18..........etc.
each term
\(\text{The formula is}\\ \boxed{~ a_n = a_{n-1} + p(n) -p(n-1) +1 \qquad n \ge 2 \qquad a_1 = 2 ~}\\ \begin{array}{rclcl} a_1 && &=& 2 \\ a_2 &=& a_1 + p(2)-p(1) + 1 = 2 + p(2)-2 + 1 &=& p(2) + 1\\ a_3 &=& a_2 + p(3)-p(2) + 1 = (~p(2) + 1~) + p(3)-p(2) + 1 &=& p(3) + 2 \\ a_4 &=& a_3 + p(4)-p(3) + 1 = (~p(3) + 2~) + p(4)-p(3) + 1 &=& p(4) + 3 \\ a_5 &=& a_4 + p(5)-p(4) + 1 = (~p(4) + 3~) + p(5)-p(4) + 1 &=& p(5) + 4\\ \cdots \\ a_n &=& a_{n-1} + p(n)-p(n-1) + 1 = (~p(n-1)+ n-2 ~)+ p(n)-p(n-1) + 1 &=& p(n) + n - 1\\ \end{array}\\ \boxed{~ a_n = n + p(n) -1 ~}\)
