[PPT] - A Stern Introduction to Combinatorial Number Theory Oliver Pechenik PowerPoint Presentation

SLIDE 1

A Stern Introduction to Combinatorial Number Theory

Oliver Pechenik February 10, 2012

Oliver Pechenik A Stern Introduction to Combinatorial Number Theory

SLIDE 2

The Stern sequence

s(1423543) = 6699.

Oliver Pechenik A Stern Introduction to Combinatorial Number Theory

SLIDE 3

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

SLIDE 4

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

SLIDE 5

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

SLIDE 6

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

SLIDE 7

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

SLIDE 8

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

SLIDE 9

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

SLIDE 10

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

SLIDE 11

Partitioning N

Let P(n) = min{sk(n) : k ∈ Z+, sk(n) = 1}.

Oliver Pechenik A Stern Introduction to Combinatorial Number Theory

SLIDE 12

Partitioning N

Let P(n) = min{sk(n) : k ∈ Z+, sk(n) = 1}. This is always a power of 2, so maybe it’s nicer to look at P2(n) := lg P(n). The function P2 : N → N is a surjection.

Oliver Pechenik A Stern Introduction to Combinatorial Number Theory

SLIDE 13

Partitioning N

Let P(n) = min{sk(n) : k ∈ Z+, sk(n) = 1}. This is always a power of 2, so maybe it’s nicer to look at P2(n) := lg P(n). The function P2 : 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 P2(n).

Oliver Pechenik A Stern Introduction to Combinatorial Number Theory

SLIDE 14

Partitioning N

Let P(n) = min{sk(n) : k ∈ Z+, sk(n) = 1}. This is always a power of 2, so maybe it’s nicer to look at P2(n) := lg P(n). The function P2 : 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 P2(n). Let O(2m) = O2(m) = {n ∈ N : P(n) = 2m}.

Oliver Pechenik A Stern Introduction to Combinatorial Number Theory

SLIDE 15

Density of O(2)

I want to understand the densities of these sets.

Oliver Pechenik A Stern Introduction to Combinatorial Number Theory

SLIDE 16

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

SLIDE 17

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

SLIDE 18

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

SLIDE 19

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

SLIDE 20

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

SLIDE 21

Density of O(2)

2e5 4e5 6e5 8e5 1e6 0.1 0.2 0.3 0.4 0.5 0.6 0.7

Oliver Pechenik A Stern Introduction to Combinatorial Number Theory

SLIDE 22

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

SLIDE 23

Log Log Plot

2 4 6 8 10 12

8
6
4
2

Oliver Pechenik A Stern Introduction to Combinatorial Number Theory

SLIDE 24

Density of O(4)

2e5 4e5 6e5 8e5 1e6 0.05 0.1 0.15

Oliver Pechenik A Stern Introduction to Combinatorial Number Theory

SLIDE 25

Density of O(8)

2e5 4e5 6e5 8e5 1e6 0.01 0.02 0.03 0.04 0.05 0.06 0.07

Oliver Pechenik A Stern Introduction to Combinatorial Number Theory

SLIDE 26

Density of O(16)

2e5 4e5 6e5 8e5 1e6 0.005 0.01 0.015 0.02 0.025

Oliver Pechenik A Stern Introduction to Combinatorial Number Theory

SLIDE 27

Density of O(32)

2e5 4e5 6e5 8e5 1e6 0.002 0.004 0.006 0.008 0.01 0.012

Oliver Pechenik A Stern Introduction to Combinatorial Number Theory

SLIDE 28

Density of O(64)

2e5 4e5 6e5 8e5 1e6 0.002 0.004 0.006 0.008 0.01

Oliver Pechenik A Stern Introduction to Combinatorial Number Theory

SLIDE 29

Density of O(128)

2e5 4e5 6e5 8e5 1e6 5e-4 1e-3 1.5e-3 2e-3 2.5e-3 3e-3 3.5e-3

Oliver Pechenik A Stern Introduction to Combinatorial Number Theory

SLIDE 30

Density of O(256)

2e5 4e5 6e5 8e5 1e6 5e-4 1e-3 1.5e-3 2e-3

Oliver Pechenik A Stern Introduction to Combinatorial Number Theory

SLIDE 31

Density of O(512)

2e5 4e5 6e5 8e5 1e6 2e-4 4e-4 6e-4 8e-4 1e-3

Oliver Pechenik A Stern Introduction to Combinatorial Number Theory

SLIDE 32

Density of O(1024)

2e5 4e5 6e5 8e5 1e6 1e-4 2e-4 3e-4 4e-4 5e-4

Oliver Pechenik A Stern Introduction to Combinatorial Number Theory

SLIDE 33

Density of O(2048)

2e5 4e5 6e5 8e5 1e6 5e-5 1e-4 1.5e-4 2e-4 2.5e-4 3e-4

Oliver Pechenik A Stern Introduction to Combinatorial Number Theory

SLIDE 34

Density of O(4096)

2e5 4e5 6e5 8e5 1e6 5e-5 1e-4 1.5e-4 2e-4

Oliver Pechenik A Stern Introduction to Combinatorial Number Theory

SLIDE 35

A Stern Introduction to Combinatorial Number Theory

Oliver Pechenik February 10, 2012

The Stern sequence

s(1423543) = 6699.

The Stern sequence

s(1423543) = 6699. We can think of the Stern sequence as a function s : N → N.

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

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

s(274) = 19

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

s(274) = 19 s(19) = 7

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

s(274) = 19 s(19) = 7 s(7) = 3

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

s(274) = 19 s(19) = 7 s(7) = 3 s(3) = 2

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

s(274) = 19 s(19) = 7 s(7) = 3 s(3) = 2 s(2) = 1

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

s(274) = 19 s(19) = 7 s(7) = 3 s(3) = 2 s(2) = 1 s(1) = 1, . . .

Partitioning N

Let P(n) = min{sk(n) : k ∈ Z+, sk(n) = 1}.

Partitioning N

Let P(n) = min{sk(n) : k ∈ Z+, sk(n) = 1}. This is always a power of 2, so maybe it’s nicer to look at P2(n) := lg P(n). The function P2 : N → N is a surjection.

Partitioning N

Let P(n) = min{sk(n) : k ∈ Z+, sk(n) = 1}. This is always a power of 2, so maybe it’s nicer to look at P2(n) := lg P(n). The function P2 : 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 P2(n).

Partitioning N

Density of O(2)

I want to understand the densities of these sets.

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?

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.

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 → ∞?

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?

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?

Density of O(2)

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

Log Log Plot

Density of O(4)

Density of O(8)

Density of O(16)

Density of O(32)

Density of O(64)

Density of O(128)

Density of O(256)

Density of O(512)

Density of O(1024)

Density of O(2048)

Density of O(4096)

Related questions

How many steps does it take on average to get to 1?

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 sk(n) = 3 for some k?