Sign up ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

Related to Gerhard's question about ascii plots. On the SeqFan mailing list was suggested to plot an integer sequence this way:

Let $F(x,y)= (x+y) (x+y+1)/2+y$ be the Cantor pairing. To plot an integer sequence $a(n)$, for a point $(x,y)$ compute $a(F(x,y))$ and assign color to the integer, e.g. in grayscale smaller is darker, for RGB/HSV there are other choices to map to color.

When $a(n)=\sigma_0(n)$ where $\sigma_0(n)$ is the number of divisors of $n$, the 2D plot shows some structure (hopefully not caused by visual artifacts).

Is there an explanation for the structure in the plot?

Color plot of $\sigma_0(F(x,y))$, smaller is darker (grayscale is quite similar):

sigma_0 and cantor pairing

When examining the integer values there are some large diagonals indeed.

share|cite|improve this question

1 Answer 1

up vote 2 down vote accepted

Have you tried to find an explanation?

The diagonals correspond to numbers $F(x,x+j).$ Every eighth one has $F(x,x+8k)=2x(x+1)+4(8k^2+4xk+3k).$ Since these are all multiples of $4$ that is already a boost.

$F(x,x+1)=2(x+1)^2.$ This is the case $q=0$ of $F(x,x-(q^2-1))=2(x-\binom{q}{2}+1-q)(x-\binom{q}{2}+1).$ So these are all pretty composite and every $q$th member is a multiple of $2q^2.$ Probably you can prove that there are no other diagonals which factor algebraically.

You will find the horizontal lines $F(\binom{j}{2}-1,y)$ worth examining.

Your image also shows possible anti-diagonals $F(x,k-x)$ but I will leave that for someone else to examine (I did not immediately see anything).

A few later comments: Along any line (the ones easily seen are horizontal, vertical and slope $\pm 1$) the values are periodic $\mod p.$ Certain dark lines can be explained by verifying that no member can divide by a small prime. I seem to recall that the lines $F(x,x-(q^2-3))$ contain no multiples of $2,3,5$ and in some cases no multiples of any prime under $30$. Some of this shows in the graphic and some not as much.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.