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A Stern Introduction to Combinatorial Number Theory Oliver Pechenik February 10, 2012 Oliver Pechenik A Stern Introduction to Combinatorial Number Theory

The Stern sequence s (1423543) = 6699. Oliver Pechenik A Stern Introduction to Combinatorial Number Theory

The Stern sequence s (1423543) = 6699. We can think of the Stern sequence as a function s : N → N . Oliver Pechenik A Stern Introduction to Combinatorial Number Theory

The Stern sequence s (1423543) = 6699. We can think of the Stern sequence as a function s : N → N . Therefore we can iterate that function and look at what happens. s (6699) = 274 Oliver Pechenik A Stern Introduction to Combinatorial Number Theory

The Stern sequence s (1423543) = 6699. We can think of the Stern sequence as a function s : N → N . Therefore we can iterate that function and look at what happens. s (6699) = 274 s (274) = 19 Oliver Pechenik A Stern Introduction to Combinatorial Number Theory

The Stern sequence s (1423543) = 6699. We can think of the Stern sequence as a function s : N → N . Therefore we can iterate that function and look at what happens. s (6699) = 274 s (274) = 19 s (19) = 7 Oliver Pechenik A Stern Introduction to Combinatorial Number Theory

The Stern sequence s (1423543) = 6699. We can think of the Stern sequence as a function s : N → N . Therefore we can iterate that function and look at what happens. s (6699) = 274 s (274) = 19 s (19) = 7 s (7) = 3 Oliver Pechenik A Stern Introduction to Combinatorial Number Theory

The Stern sequence s (1423543) = 6699. We can think of the Stern sequence as a function s : N → N . Therefore we can iterate that function and look at what happens. s (6699) = 274 s (274) = 19 s (19) = 7 s (7) = 3 s (3) = 2 Oliver Pechenik A Stern Introduction to Combinatorial Number Theory

The Stern sequence s (1423543) = 6699. We can think of the Stern sequence as a function s : N → N . Therefore we can iterate that function and look at what happens. s (6699) = 274 s (274) = 19 s (19) = 7 s (7) = 3 s (3) = 2 s (2) = 1 Oliver Pechenik A Stern Introduction to Combinatorial Number Theory

The Stern sequence s (1423543) = 6699. We can think of the Stern sequence as a function s : N → N . Therefore we can iterate that function and look at what happens. s (6699) = 274 s (274) = 19 s (19) = 7 s (7) = 3 s (3) = 2 s (2) = 1 s (1) = 1 , . . . Oliver Pechenik A Stern Introduction to Combinatorial Number Theory

Partitioning N Let P ( n ) = min { s k ( n ) : k ∈ Z + , s k ( n ) � = 1 } . Oliver Pechenik A Stern Introduction to Combinatorial Number Theory

Partitioning N Let P ( n ) = min { s k ( n ) : k ∈ Z + , s k ( n ) � = 1 } . This is always a power of 2, so maybe it’s nicer to look at P 2 ( n ) := lg P ( n ). The function P 2 : N → N is a surjection. Oliver Pechenik A Stern Introduction to Combinatorial Number Theory

Partitioning N Let P ( n ) = min { s k ( n ) : k ∈ Z + , s k ( n ) � = 1 } . This is always a power of 2, so maybe it’s nicer to look at P 2 ( n ) := lg P ( n ). The function P 2 : N → N is a surjection. Either way we get a partition of N into an infinite number of parts according to the values of P ( n ) or P 2 ( n ). Oliver Pechenik A Stern Introduction to Combinatorial Number Theory

Partitioning N Let P ( n ) = min { s k ( n ) : k ∈ Z + , s k ( n ) � = 1 } . This is always a power of 2, so maybe it’s nicer to look at P 2 ( n ) := lg P ( n ). The function P 2 : N → N is a surjection. Either way we get a partition of N into an infinite number of parts according to the values of P ( n ) or P 2 ( n ). Let O (2 m ) = O 2 ( m ) = { n ∈ N : P ( n ) = 2 m } . Oliver Pechenik A Stern Introduction to Combinatorial Number Theory

Density of O (2) I want to understand the densities of these sets. Oliver Pechenik A Stern Introduction to Combinatorial Number Theory

Density of O (2) I want to understand the densities of these sets. Given a “random” integer n , what is the probability that P ( n ) = 2? Oliver Pechenik A Stern Introduction to Combinatorial Number Theory

Density of O (2) I want to understand the densities of these sets. Given a “random” integer n , what is the probability that P ( n ) = 2? Can compute | O (2) ∩ [ n ] | / n for large n . Oliver Pechenik A Stern Introduction to Combinatorial Number Theory

Density of O (2) I want to understand the densities of these sets. Given a “random” integer n , what is the probability that P ( n ) = 2? Can compute | O (2) ∩ [ n ] | / n for large n . Does this approach a limit as n → ∞ ? Oliver Pechenik A Stern Introduction to Combinatorial Number Theory

Density of O (2) I want to understand the densities of these sets. Given a “random” integer n , what is the probability that P ( n ) = 2? Can compute | O (2) ∩ [ n ] | / n for large n . Does this approach a limit as n → ∞ ? If so, what limit? Oliver Pechenik A Stern Introduction to Combinatorial Number Theory

Density of O (2) I want to understand the densities of these sets. Given a “random” integer n , what is the probability that P ( n ) = 2? Can compute | O (2) ∩ [ n ] | / n for large n . Does this approach a limit as n → ∞ ? If so, what limit? Is it bounded away from 0? Away from 1? Oliver Pechenik A Stern Introduction to Combinatorial Number Theory

Density of O (2) 0.7 0.6 0.5 0.4 0.3 0.2 0.1 2e5 4e5 6e5 8e5 1e6 Oliver Pechenik A Stern Introduction to Combinatorial Number Theory

Densities in the first million integers 2 0.744690 4 0.156660 8 0.063105 16 0.018228 32 0.006627 64 0.006991 128 0.002295 256 0.000424 512 0.000401 1024 0.000338 2048 0.000160 4096 0.000068 Oliver Pechenik A Stern Introduction to Combinatorial Number Theory

Log Log Plot 2 4 6 8 10 12 -2 -4 -6 -8 Oliver Pechenik A Stern Introduction to Combinatorial Number Theory

Density of O (4) 0.15 0.1 0.05 2e5 4e5 6e5 8e5 1e6 Oliver Pechenik A Stern Introduction to Combinatorial Number Theory

Density of O (8) 0.07 0.06 0.05 0.04 0.03 0.02 0.01 2e5 4e5 6e5 8e5 1e6 Oliver Pechenik A Stern Introduction to Combinatorial Number Theory

Density of O (16) 0.025 0.02 0.015 0.01 0.005 2e5 4e5 6e5 8e5 1e6 Oliver Pechenik A Stern Introduction to Combinatorial Number Theory

Density of O (32) 0.012 0.01 0.008 0.006 0.004 0.002 2e5 4e5 6e5 8e5 1e6 Oliver Pechenik A Stern Introduction to Combinatorial Number Theory

Density of O (64) 0.01 0.008 0.006 0.004 0.002 2e5 4e5 6e5 8e5 1e6 Oliver Pechenik A Stern Introduction to Combinatorial Number Theory

Density of O (128) 3.5e-3 3e-3 2.5e-3 2e-3 1.5e-3 1e-3 5e-4 2e5 4e5 6e5 8e5 1e6 Oliver Pechenik A Stern Introduction to Combinatorial Number Theory

Density of O (256) 2e-3 1.5e-3 1e-3 5e-4 2e5 4e5 6e5 8e5 1e6 Oliver Pechenik A Stern Introduction to Combinatorial Number Theory

Density of O (512) 1e-3 8e-4 6e-4 4e-4 2e-4 2e5 4e5 6e5 8e5 1e6 Oliver Pechenik A Stern Introduction to Combinatorial Number Theory

Density of O (1024) 5e-4 4e-4 3e-4 2e-4 1e-4 2e5 4e5 6e5 8e5 1e6 Oliver Pechenik A Stern Introduction to Combinatorial Number Theory

Density of O (2048) 3e-4 2.5e-4 2e-4 1.5e-4 1e-4 5e-5 2e5 4e5 6e5 8e5 1e6 Oliver Pechenik A Stern Introduction to Combinatorial Number Theory

Density of O (4096) 2e-4 1.5e-4 1e-4 5e-5 2e5 4e5 6e5 8e5 1e6 Oliver Pechenik A Stern Introduction to Combinatorial Number Theory

Related questions How many steps does it take on average to get to 1? Oliver Pechenik A Stern Introduction to Combinatorial Number Theory

Related questions How many steps does it take on average to get to 1? It is also possible to look at “watersheds” for numbers that are not powers of 2. What is the probability that s k ( n ) = 3 for some k ? Oliver Pechenik A Stern Introduction to Combinatorial Number Theory