Fibonacci

[Emeric Deutsch]

Hello Yasutoshi,
Here are the first 11 terms of your first sequence:
2, 3, 7, 23, 103, 613, 4751, 47137, 582511, 8758339, 156819893
I have used Maple.
First I wrote a program to get InversePrime (not necessarily the
best):
ip:=proc(n) local P, i: P:={}: if isprime(n)=true then for i from 2 to n 
do if isprime(i)=true then P:=P union {i} else P:=P fi: od: nops(P) else X 
fi: end;

Then sequence is given by

a:=proc(n) if n=1 then 2 elif n=2 then 3 else ithprime(ip(a(n-2))+a(n-1)) 
fi end; seq(a(n),n=1..9);

First 9 terms are obtained very fast.

Emeric