Numerators of harmonic numbers

Harmonic numbers

The nth harmonic number, Hn, is the sum of the reciprocals of the integers up to and including n. For example,

H4 = 1 + 1/2 + 1/3 + 1/4 = 25/12.

Here’s a curious fact about harmonic numbers, known as Wolstenholme’s theorem:

For a prime p > 3, the numerator of Hp-1 is divisible by p2.

The example above shows this for p = 5. In that case, the numerator is not just divisible by p2, it is p2, though this doesn’t hold in general. For example, H10 = 7381/2520. The numerator 7381 is divisible by 112 = 121, but it’s not equal to 121.

Generalized harmonic numbers

The generalized harmonic numbers Hn,m are the sums of the reciprocals of the first n positive integers, each raised to the power m. Wolstenholme’s theorem also says something about these numbers too:

For a prime p > 3, the numerator of Hp-1,2 is divisible by p.

For example, H4,2 = 205/144, and the numerator is clearly divisible by 5.

Computing with Python

You can play with harmonic numbers and generalized harmonic numbers in Python using SymPy. Start with the import statement

from sympy.functions.combinatorial.numbers import harmonic

Then you can get the nth harmonic number with harmonic(n) and generalized harmonic numbers with harmonic(n, m).

To extract the numerators, you can use the method as_numer_denom to turn the fractions into (numerator, denominator) pairs. For example, you can create a list of the denominators of the first 10 harmonic numbers with

[harmonic(n).as_numer_denom()[0] for n in range(10)]

What about 0?

You might notice that harmonic(0) returns 0, as it should. The sum defining the harmonic numbers is empty in this case, and empty sums are defined to be zero.

Scientific computing in Python

Scientific computing in Python is expanding and maturing rapidly. Last week at the SciPy 2015 conference there were about twice as many people as when I’d last gone to the conference in 2013.

You can get some idea of the rapid develop of the scientific Python stack and its future direction by watching the final keynote of the conference by Jake VanderPlas.

I used Python for a while before I discovered that there were so many Python libraries for scientific computing. At the time I was considering learning Ruby or some other scripting language, but I committed to Python when I found out that Python has far more libraries for the kind of work I do than other languages do. It felt like I’d discovered a secret hoard of code. I expect it would be easier today to discover the scientific Python stack. (It really is becoming more of an integrated stack, not just a collection of isolated libraries. This is one of the themes in the keynote above.)

When people ask me why I use Python, rather than languages like Matlab or R, my response is that I do a mixture of mathematical programming and general programming. I’d rather do mathematics in a general programming language than do general programming in a mathematical language.

One of the drawbacks of Python, relative to C++ and related languages, is speed. This is a problem in languages like R as well. However, with Python there are ways to speed up code without completely rewriting it, such as Cython and Numba. The only reliable way I’ve found to make R much faster, is to rewrite it in another language.

Another drawback of Python until recently was that data manipulation and exploration were not as convenient as one would hope. However, that has changed due to developments such as Pandas, initiated by Wes McKinney. For more on how that came to be and where it’s going, see his keynote from the second day of SciPy 2015.

It’s not clear why Python has become the language of choice for so many people in scientific computing. Maybe if people like Travis Oliphant had decided to use some other language for scientific programming years ado, we’d all be using that language now. Python wasn’t intended to be a scientific programming language. And as Jake VanderPlas points out in his keynote, Python still is not a scientific programming language, but the foundation for a scientific programming stack. Maybe Python’s strength is that it’s not a scientific language. It has drawn more computer scientists to contribute to the core language than it would have if it had been more of a domain-specific language.

* * *

If you’d like help moving to the Python stack, please let me know.

Mystery curve

This afternoon I got a review copy of the book Creating Symmetry: The Artful Mathematics of Wallpaper Patterns. Here’s a striking curves from near the beginning of the book, one that the author calls the “mystery curve.”

The curve is the plot of exp(it) – exp(6it)/2 + i exp(-14it)/3 with t running from 0 to 2π.

Here’s Python code to draw the curve.

import matplotlib.pyplot as plt
from numpy import pi, exp, real, imag, linspace

def f(t):
    return exp(1j*t) - exp(6j*t)/2 + 1j*exp(-14j*t)/3

t = linspace(0, 2*pi, 1000)

plt.plot(real(f(t)), imag(f(t)))

# These two lines make the aspect ratio square
fig = plt.gcf()

Maybe there’s a more direct way to plot curves in the complex plane rather than taking real and imaginary parts.

Updated code for the aspect ratio per Janne’s suggestion in the comments.

Related posts:

Several people have been making fun visualizations that generalize the example above.

Brent Yorgey has written two posts, one choosing frequencies randomly and another that animates the path of a particle along the curve and shows how the frequency components each contribute to the motion.

Mike Croucher developed a Jupyter notebook that lets you vary the frequency components with sliders.

John Golden created visualizations in GeoGerba here and here.

Jennifer Silverman showed how these curves are related to decorative patterns that popular in the 1960’s. She also created a coloring book and a video.

Dan Anderson accused me of nerd sniping him and created this visualization.

Rotating PDF pages with Python

Yesterday I got a review copy of Automate the Boring Stuff with Python. It explains, among other things, how to manipulate PDFs from Python. This morning I needed to rotate some pages in a PDF, so I decided to try out the method in the book.

The sample code uses PyPDF2. I’m using Conda for my Python environment, and PyPDF2 isn’t directly available for Conda. I searched Binstar with

binstar search -t conda pypdf2

The first hit was from JimInCO, so I installed PyPDF2 with

conda install -c pypdf2

I scanned a few pages from a book to PDF, turning the book around every other page, so half the pages in the PDF were upside down. I needed a script to rotate the even numbered pages. The script counts pages from 0, so it rotates the odd numbered pages from its perspective.

import PyPDF2

pdf_in = open('original.pdf', 'rb')
pdf_reader = PyPDF2.PdfFileReader(pdf_in)
pdf_writer = PyPDF2.PdfFileWriter()

for pagenum in range(pdf_reader.numPages):
    page = pdf_reader.getPage(pagenum)
    if pagenum % 2:

pdf_out = open('rotated.pdf', 'wb')

It worked as advertised on the first try.

Random walks and the arcsine law

Suppose you stand at 0 and flip a fair coin. If the coin comes up heads, you take a step to the right. Otherwise you take a step to the left. How much of the time will you spend to the right of where you started?

As the number of steps N goes to infinity, the probability that the proportion of your time in positive territory is less than x approaches 2 arcsin(√x)/π. The arcsine term gives this rule its name, the arcsine law.

Here’s a little Python script to illustrate the arcsine law.

import random
from numpy import arcsin, pi, sqrt

def step():
u = random.random()
return 1 if u < 0.5 else -1 M = 1000 # outer loop N = 1000 # inner loop x = 0.3 # Use any 0 < x < 1 you'd like. outer_count = 0 for _ in range(M): n = 0 position= 0 inner_count = 0 for __ in range(N): position += step() if position > 0:
inner_count += 1
if inner_count/N < x: outer_count += 1 print (outer_count/M) print (2*arcsin(sqrt(x))/pi) [/code]

Playing with continued fractions and Khinchin’s constant

Take a real number x and expand it as a continued fraction. Compute the geometric mean of the first n coefficients.

Aleksandr Khinchin proved that for almost all real numbers x, as n → ∞ the geometric means converge. Not only that, they converge to the same constant, known as Khinchin’s constant, 2.685452001…. (“Almost all” here mean in the sense of measure theory: the set of real numbers that are exceptions to Khinchin’s theorem have measure zero.)

To get an idea how fast this convergence is, let’s start by looking at the continued fraction expansion of π. In Sage, we can type


to get the continued fraction coefficient

[3, 7, 15, 1, 292, 1, 1, 1, 2, 1, 3, 1, 14, 2, 1, 1, 2, 2, 2, 2, 1, 84, 2, 1, 1, 15, 3]

for π to 100 decimal places. The geometric mean of these coefficients is 2.84777288486, which only matches Khinchin’s constant to 1 significant figure.

Let’s try choosing random numbers and working with more decimal places.

There may be a more direct way to find geometric means in Sage, but here’s a function I wrote. It leaves off any leading zeros that would cause the geometric mean to be zero.

from numpy import exp, mean, log
def geometric_mean(x):
    return exp( mean([log(k) for k in x if k > 0]) )

Now let’s find 10 random numbers to 1,000 decimal places.

for _ in range(10):
    r = RealField(1000).random_element(0,1)

This produced


Three of these agree with Khinchin’s constant to two significant figures but the rest agree only to one. Apparently the convergence is very slow.

If we go back to π, this time looking out 10,000 decimal places, we get a little closer:


produces 2.67104567579, which differs from Khinchin’s constant by about 0.5%.

Counting primitive bit strings

A string of bits is called primitive if it is not the repetition of several copies of a smaller string of bits. For example, the 101101 is not primitive because it can be broken down into two copies of the string 101. In Python notation, you could produce 101101 by "101"*2. The string 11001101, on the other hand, is primitive. (It contains substrings that are not primitive, but the string as a whole cannot be factored into multiple copies of a single string.)

For a given n, let’s count how many primitive bit strings there are of length n. Call this f(n). There are 2n bit strings of length n, and f(n) of these are primitive. For example, there are f(12) primitive bit strings of length 12. The strings that are not primitive are made of copies of primitive strings: two copies of a primitive string of length 6, three copies of a primitive string of length 4, etc. This says

 2^{12} = f(12) + f(6) + f(4) + f(3) + f(2) + f(1)

and in general

2^n = \sum_{d \mid n} f(d)

Here the sum is over all positive integers d that divide n.

Unfortunately this formula is backward. It gives is a formula for something well known, 2n, as a sum of things we’re trying to calculate. The Möbius inversion formula is just what we need to turn this formula around so that the new thing is on the left and sums of old things are on the right. It tells us that

f(n) = \sum_{d \mid n} \mu\left(\frac{n}{d}\right) 2^d

where μ is the Möbius function.

We could compute f(n) with Python as follows:

from sympy.ntheory import mobius, divisors

def num_primitive(n):
    return sum( [mobius(n/d)*2**d for d in divisors(n)] )

The latest version of SymPy, version 0.7.6, comes with a function mobius for computing the Möbius function. If you’re using an earlier version of SymPy, you can roll your own mobius function:

from sympy.ntheory import factorint

def mobius(n):
    exponents = factorint(n).values()
    lenexp = len(exponents)
    m = 0 if lenexp == 0 else max(exponents)
    return 0 if m > 1 else (-1)**lenexp

The version of mobius that comes with SymPy 0.7.6 may be more efficient. It could, for example, stop the factorization process early if it discovers a square factor.

Ellipsoid surface area

How much difference does the earth’s equatorial bulge make in its surface area?

To first approximation, the earth is a sphere. The next step in sophistication is to model the earth as an ellipsoid.

The surface area of an ellipsoid with semi-axes abc is

A = 2\pi \left( c^2 + \frac{ab}{\sin\phi} \left( E(\phi, k) \sin^2\phi + F(\phi, k) \cos^2 \phi\right)\right)




m = k^2 = \frac{a^2(b^2 - c^2)}{b^2(a^2 - c^2)}

The functions E and F are incomplete elliptic integrals

 F(\phi, k) = \int_0^\phi \frac{d\theta}{\sqrt{1 - k^2 \sin^2\theta}}


E(\phi, k) = \int_0^\phi \sqrt{1 - k^2 \sin^2\theta}\,d\theta

implemented in SciPy as ellipeinc and ellipkinc. Note that the SciPy functions take m as their second argument rather its square root k.

For the earth, a = b and so m = 1.

The following Python code computes the ratio of earth’s surface area as an ellipsoid to its area as a sphere.

from scipy import pi, sin, cos, arccos
from scipy.special import ellipkinc, ellipeinc

# values in meters based on GRS 80
equatorial_radius = 6378137
polar_radius = 6356752.314140347

a = b = equatorial_radius
c = polar_radius

phi = arccos(c/a)
# in general, m = (a**2 * (b**2 - c**2)) / (b**2 * (a**2 - c**2))
m = 1

temp = ellipeinc(phi, m)*sin(phi)**2 + ellipkinc(phi, m)*cos(phi)**2
ellipsoid_area = 2*pi*(c**2 + a*b*temp/sin(phi))

# sphere with radius equal to average of polar and equatorial
r = 0.5*(a+c)
sphere_area = 4*pi*r**2


This shows that the ellipsoid model leads to 0.112% more surface area relative to a sphere.

Source: See equation 19.33.2 here.

Update: It was suggested in the comments that it would be better to compare the ellipsoid area to that of a sphere of the same volume. So instead of using the average of the polar and equatorial radii, one would take the geometric mean of the polar radius and two copies of the equatorial radius. Using that radius, the ellipsoid has 0.0002% more area than the sphere.

Benchmarking C++, Python, R, etc.

The other day Travis Oliphant pointed out an interesting paper: A Comparison of Programming Languages in Economics. The paper benchmarks several programming languages on a computational problem in economics.

All the usual disclaimers about benchmarks apply, your mileage may vary, etc. See the paper for details.

Here I give my summary of their summary of their results. The authors ran separate benchmarks on Mac and Windows. The results were qualitatively the same, so I just report the Windows results here.

Times in the table below are relative to the fastest C++ run.

Language Time
C++ 1.00
Java 2.10
Julia 2.70
CPython 155.31
Python with Numba 1.57
R 505.09
R using compiler package 243.38


The most striking result is that the authors were able to run their Python code 100x faster, achieving performance comparable to C++, by using Numba.

Number theory determinant and SymPy

Let σ(n) be the sum of the positive divisors of n and let gcd(a, b) be the greatest common divisor of a and b.

Form an n by n matrix M whose (i, j) entry is σ(gcd(i, j)). Then the determinant of M is n!.

The following code shows that the theorem is true for a few values of n and shows how to do some common number theory calculations in SymPy.

from sympy import gcd, divisors, Matrix, factorial

def f(i, j):
    return sum( divisors( gcd(i, j) ) )

def test(n):
    r = range(1, n+1)
    M = Matrix( [ [f(i, j) for j in r] for i in r] )
    return M.det() - factorial(n)

for n in range(1, 11):
    print test(n)

As expected, the test function returns zeros.

If we replace the function σ above by τ where τ(n) is the number of positive divisors of n, the corresponding determinant is 1. To test this, replace sum by len in the definition of f and replace factorial(n) by 1.

In case you’re curious, both results are special cases of the following more general theorem. I don’t know whose theorem it is. I found it here.

For any arithmetic function f(m), let g(m) be defined for all positive integers m by

g(m) = \sum_{d \,\mid \,m} \mu(d) f\left(\frac{m}{d}\right)

Let M be the square matrix of order n with ij element f(gcd(i, j)). Then

\det M = \prod_i^n g(j)

Here μ is the Möbius function. The two special cases above correspond to g(m) = m and g(m) = 1.

A prime-generating formula and SymPy

Mills’ constant is a number θ such that the integer part of θ raised to a power of 3 is always a prime. We’ll see if we can verify this computationally with SymPy.

from sympy import floor, isprime
from sympy.mpmath import mp, mpf

# set working precision to 200 decimal places
mp.dps = 200

# Mills' constant
theta = mpf("1.30637788386308069046861449260260571291678458515671364436805375996643405376682659882150140370119739570729")

for i in range(1, 7):
    n = floor(theta**3**i)
    print i, n, isprime(n)

Note that the line of code defining theta wraps Mill’s constant as a string and passes it as an argument to mpf. If we just wrote theta = 1.30637788386308… then theta would be truncated to an ordinary floating point number and most of the digits would be lost. Not that I did that in the first version of the code. :)

This code says that the first five outputs are prime but that the sixth is not. That’s because we are not using Mills’ constant to enough precision. The integer part of θ^3^6 is 85 digits long, and we don’t have enough precision to compute the last digit correctly.

If we use the value of Mills’ constant given here to 640 decimal places, and increase mp.dps to 1000, we can correctly compute the sixth term, but not the seventh.

Mill’s formula is just a curiosity with no practical use that I’m aware of, except possibly as an illustration of impractical formulas.

Relating Airy and Bessel functions

The Airy functions Ai(x) and Bi(x) are independent solutions to the differential equation

y'' - xy = 0

For negative x they act something like sin(x) and cos(x). For positive x they act something like exp(x) and exp(-x). This isn’t surprising if you look at the differential equation. If you replace x with a negative constant, you get sines and cosines, and if you replace it with a positive constant, you get positive and negative exponentials.

The Airy functions can be related to Bessel functions as follows:

\mathrm{Ai}(x) = \left\{ 	\begin{array}{ll} 		\frac{1}{3}\sqrt{\phantom{-}x} \left(I_{-1/3}(\hat{x}) - I_{1/3}(\hat{x})\right) & \mbox{if } x > 0 \\<br /><br /><br /><br />
                \\<br /><br /><br /><br />
		\frac{1}{3}\sqrt{-x} \left(J_{-1/3}(\hat{x}) + J_{1/3}(\hat{x})\right) & \mbox{if } x < 0 	\end{array} \right.


\mathrm{Bi}(x) = \left\{ 	\begin{array}{ll} 		\sqrt{\phantom{-}x/3} \left(I_{-1/3}(\hat{x}) + I_{1/3}(\hat{x})\right) & \mbox{if } x > 0 \\<br />                 \\<br /> 		\sqrt{-x/3} \left(J_{-1/3}(\hat{x}) - J_{1/3}(\hat{x})\right) & \mbox{if } x < 0 	\end{array} \right.

Here J is a “Bessel function of the first kind” and I is a “modified Bessel function of the first kind.” Also

\hat{x} = \frac{2}{3} \left(\sqrt{|x|}\right)^3

To verify the equations above, and to show how to compute these functions in Python, here’s some code.

The SciPy function airy computes both functions, and their first derivatives, at once. I assume that’s because it doesn’t take much longer to compute all four functions than to compute one. The code for Ai2 and Bi2 below uses np.where instead of if ... else so that it can operate on NumPy vectors all at once. You can plot Ai and Ai2 and see that the two curves lie on top of each other. The same holds for Bi and Bi2.


from scipy.special import airy, jv, iv
from numpy import sqrt, where

def Ai(x):
    (ai, ai_prime, bi, bi_prime) = airy(x)
    return ai

def Bi(x):
    (ai, ai_prime, bi, bi_prime) = airy(x)
    return bi

def Ai2(x):
    third = 1.0/3.0
    hatx = 2*third*(abs(x))**1.5
    return where(x > 0,
        third*sqrt( x)*(iv(-third, hatx) - iv(third, hatx)),
        third*sqrt(-x)*(jv(-third, hatx) + jv(third, hatx)))

def Bi2(x):
    third = 1.0/3.0
    hatx = 2*third*(abs(x))**1.5
    return where(x > 0,
        sqrt( x/3.0)*(iv(-third, hatx) + iv(third, hatx)),
        sqrt(-x/3.0)*(jv(-third, hatx) - jv(third, hatx)))


There is a problem with Ai2 and Bi2: they return nan at 0. A more careful implementation would avoid this problem, but that’s not necessary since these functions are only for illustration. In practice, you’d simply use airy and it does the right thing at 0.

Related links:

SymPy book

There’s a new book on SymPy, a Python library for symbolic math.

The book is Instant SymPy Starter by Ronan Lamy. As far as I know, this is the only book just on SymPy. It’s only about 50 pages, which is nice. It’s not a replacement for the online documentation but just a quick start guide.

The online SymPy documentation is good, but I think it would be easier to start with this book. And although I’ve been using SymPy off and on for a while, I learned a few things from the book.