A319864 Exponents of the final nontrivial entry of the iterated Stern sequence; a(n) = log_2 min{s^k(n) : k > 0, s^k(n) > 1}, where s(n) = A002487(n).

1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 1, 1, 1, 2, 4, 1, 2, 1, 1, 3, 1, 1, 1, 1, 1, 3, 1, 1, 2, 1, 5, 1, 1, 2, 2, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 2, 2, 1, 1, 6, 1, 1, 1, 1, 1, 2, 1, 2, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 3, 3, 1, 2, 1, 1, 1, 1, 1, 4, 2, 1, 1, 1, 2, 4, 1, 1, 1, 1, 1, 2, 1, 3, 3, 1, 1

Offset: 2

Author

Oliver Pechenik, Sep 29 2018

Keywords

• nonn

Comments

Let s(n) = A002487(n). Since s(n) < n for n > 1, iterating A002487 from any starting point eventually yields the fixed point 1 = s(1). Since s^-1(1) consists of the powers of 2, min{s^k(n) : k > 0, s^k(n) > 1} is a nontrivial power of 2 for any n > 1. Hence, the entries of this sequence are integers.

Since a(2^m) = m, every positive integer appears in this sequence.

Question: What is the asymptotic density of the number 1 in this sequence? Of the first 10^6 entries, more than 74% are 1.

Examples

			Letting s(m) = A002487(m), we have s(7) = 3, s(3) = 2, and s(2) = 1. Hence, a(7) = log_2(2) = 1.
		

Crossrefs

Cf. A002487.

Programs

Mathematica

s[n_] := If[n<2, n, If[EvenQ[n], s[n/2], s[(n-1)/2] + s[(n+1)/2]]];  a[n_] := Module[{nn = s[n]}, If[nn==1, Log2[n], a[nn]]]; Array[a, 100, 2] (* Amiram Eldar, Nov 22 2018 *)

PARI

s(n) = if( n<2, n>0, s(n\2) + if( n%2, s(n\2 + 1))); \\ A002487
a(n) = while((nn = s(n)) != 1, n = nn); valuation(n, 2); \\ Michel Marcus, Nov 23 2018

Python

from math import log
def s(n): return n if n<2 else s(n//2) if n%2==0 else s((n-1)//2)+s((n+1)//2)
def a(n): nn = s(n); return int(log(n,2)) if nn==1 else a(nn)
print([a(n) for n in range(2, 100)])

Python

from functools import reduce
def A319864(n):
    while (m:=sum(reduce(lambda x,y:(x[0],x[0]+x[1]) if int(y) else (x[0]+x[1],x[1]),bin(n)[-1:2:-1],(1,0)))) > 1:
        n = m
    return n.bit_length()-1 # Chai Wah Wu, May 18 2023