quantaum mathematics chapter 9 : oscillating number 130

Chapter 9  oscillating number

as you know about indium function which is a additivie function

ex:  i(10)=i(5)+i(2)=p(5)+p(2)=11+3=14

now if you repeatedly apply then you get a prime number after it goes on incresing. for example

i(38)=i(19)+i(2)=67+3=70

i(i(38))=i(70)=i(7)+i(5)+i(2)=17+11+3=31=i^2(38)

i( i(i(38)))=i(31)=127

after applying it the number keep on increasing but there are certain numbers in which it don't happen

such numbers are called oscillating number

i^n(m)=m 

then m is oscillating number n is natural number also n is called frequency. first oscillating number is 130 in which frequency is 7.

                                                        

i(130)=i(2)+i(5)+i(13)=3+11+41=55

i^2(130)=i(55)=i(5)+i(11)=42

i^3(130)=i(42)=i(7)+i(2)+i(3)=17+3+5=25

i^4(130)=i(25)=2*i(5)=2*11=22

i^5(130)=i(22)=i(11)+i(2)=34

i^6(130)=i(34)=i(17)+i(2)=62

i^7(130)=i(62)=i(31)+i(2)=130

also number that occurs are also oscillating for example

22,25,34,42,55,62  are partially oscillating number with 130