[seqfan] A monotonically increasing sequence with a(a(a(n))) being odd

[Eric Angelini]

Hello SeqFans,

The seq starts :

S = 1,3,4,5,7,8,9,10,11,13,15,16,17,18,19,20,21,22,23,25,27,29,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,49,51,53,55,57,59,61,63,64,65,66,...

Looking at runs of consecutive integers we see that we have
(after the first integer, "1"):

1st run length is  3 (= 2^1 + 1)    3,4,5,      
2nd run length is  5 (= 2^2 + 1)    7,8,9,10,11,   
3rd run length is  9 (= 2^3 + 1)    15,16,17,18,19,20,21,22,23,   
4th run length is 17 (= 2^4 + 1)    31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,
... etc.

Those runs are separated by runs of consecutive odd integers:

1st run length is 0 (= 2^0 - 1)    -
2nd run length is 1 (= 2^1 - 1)    13
3rd run length is 3 (= 2^2 - 1)    25,27,29,
4th run length is 7 (= 2^3 - 1)    49,51,53,55,57,59,61,
... etc.

So a general formula for a(a(a(n))) is not impossible to find.

Missing terms in S are:

M(s) = 2,6,12,14,24,26,28,30,48,50,52,54,56,58,60,62,...

---

I'm curious to see a monotonically increasing sequence with 
a(a(a(n))) being prime.

Best,
É.