OFFSET
1,2
COMMENTS
Theorem 1: Every positive numbers appears at least once.
Proof from Keith F. Lynch, Jan 04 2020:
Since no nonzero power of 2 equals a power of 10, i.e., log(10)/log(2) is irrational, any sequence in which each term is the double of the previous term will start with every decimal number infinitely many times. So any S will terminate after a finite number of steps, and the next missing number will be used as S. QED
Theorem 2: No term is repeated.
Proof:
Suppose N is repeated, so there are a pair of chains
{S, 2*S, 4*S, ..., N = 2^i*S, ...},
{T, 2*T, 4*T, ..., N = 2^j*T, ...},
where T occurs after S. There are two cases. If i>=j then T = 2^(i-j)*S, so T was not a missing number. If i<j then S = 2^(j-i)*T, and T would have been a smaller choice for S. QED
So this is a permutation of the positive integers.
From Rémy Sigrist, Jan 23 2020: (Start)
The sequence can naturally be seen as an irregular table where:
- T(n, k+1) = 2*T(n, k) for k = 1..A331442(n)-1.
(End)
REFERENCES
Eric Angelini, Posting to Math Fun Mailing List, Jan 04 2020.
LINKS
Rémy Sigrist, Table of n, a(n) for n = 1..10103 (first 168 chains)
Rémy Sigrist, PARI program for A331440
EXAMPLE
The process begins like this:
Initially S = 1 is the smallest missing number, so we have:
S = 1, 2, 4, 8, 16, stop (because 16 contains S), S = 3, 6, 12, 24, 48, 96, 192, 384, stop, S = 5, 10, 20, 40, 80, 60, 320, 640, 1280, 2560, stop, S = 7, 14, 28, 56, 112, 224, 448, 896, 1792, stop, S = 9, 18, 36, 72, ...
PROG
(PARI) See Links section.
CROSSREFS
KEYWORD
nonn,base,tabf
AUTHOR
N. J. A. Sloane, Jan 21 2020
STATUS
approved