fun1=e^(0.39513 x + Cos[29.45144 + 0.3697 x])

fun2=e^(0.96964 x - 1.47 Cos[1.038 x])/Fibonacci[1.214 x - 2.3775 Cos[1.0423 x]]



ff[x_] = If[x < 11, If[x > 5, (fun1 + fun2)/2, fun1],
If[x < 25, fun2, (fun1 + fun2)/2]]
If[x < 11, If[x > 5, (fun1 + fun2)/2, fun1],
If[x < 25, fun2, (fun1 + fun2)/2]]
{{1, 2}, {3, 3}, {6, 5}, {11, 7}, {19, 13}}
{{22, 17}, {31, 19}, {48, 31}, {74, 61}, {92, 89}}
{{152, 107}, {179, 127}, {285, 521}, {575, 607}, {1096, 1279}}
{{1542, 2203}, {1679, 2281}, {1898, 3217}, {2889, 4253}, {5783, 4423}}
{{11397, 9689}, {16765, 9941}, {18455, 11213}, {20173, 19937}, {31480, 21701}}
{{46762, 23209}, {77296, 44497}, {110097, 86243}, {123109,  110503}, {134596, 132049}}
{{187989, 216091}, {353504, 397873}, {725362, 859433}, {1217224, 1257787}, {1614862, 1398269}}
{{2211614, 2976221}, {3686867, 3021377}, {6937950,  6972593}, {12930869, 13466917}, {20228588, 20996011}}
{{24506612, 24036583}, {26215020, 25964951}, {30269917,  30402457}, {44211066, 32,582,657}, {80232303, ?}}
44th mp found several months later
My forecast for Mersenne primes back in
July 24, 2005
Below, ff, my forecasting formula, is graphed and mp's are appended.