Totient and Sigma Graphs Revisited

Not long ago, in March of 2025, I made a blog post titled Totient Function: Jagged Versus Rounded Local Minima about a special feature in the graph of the totient function. I termed this special feature the "rounded local minimum" as opposed to the more "jagged local minimum". I was reminded of this graphical curiousity because today the number associated with my diurnal age, 27885, marks such a rounded local minimum or "2-nadirs" as the OEIS puts it. Such numbers belong to OEIS A076773 : 

  A076773: 2-nadirs of \( \phi\): numbers \(k\) such that:

 \( \phi(k-2) > \phi(k-1) > \phi(k) < \phi(k+1) < \phi(k+2)\).

Figure 1 shows a graph of the numbers and the totients in the region of 27885:

Figure 1

In relation to the sigma or sum of divisors function, 27885 exhibits what might be called a "local rounded maximum" as opposed to a "jagged local maximum". These I also discuss in my Totient Function: Jagged Versus Rounded Local Minima blog. Figure 2 shows the situation.

Figure 2

Such numbers constitute OEIS A323380:

A323380   Odd \(k\) such that \( \sigma(k) > \sigma(k+1) \) and \( \sigma(k) > \sigma(k-1) \)

This turns out to be the same as A076773 with "sigma" replacing "phi" and "zenith" replacing "nadir":

  2-zeniths of \( \sigma\): numbers \(k\) such that:

 \( \sigma(k-2) > \sigma(k-1) > \sigma(k) < \sigma (k+1) < \sigma(k+2)\).

In the previously mentioned blog, I listed the numbers in OEIS A076773 and A323380 up to 40000 but I didn't realise at the time how many numbers these two sequences had in common. Here are the numbers that belong in both sequences (up to 40000):

315, 525, 1155, 1575, 1755, 1785, 1995, 2475, 2805, 3045, 3315, 3465, 3885, 4095, 4125, 4515, 4725, 5115, 5355, 5775, 6045, 6195, 6405, 6435, 6615, 6825, 7035, 7245, 7605, 8085, 8505, 8715, 8925, 9135, 9405, 9555, 9765, 9975, 10395, 11235, 11385, 11445, 11655, 12075, 12285, 12675, 12705, 12915, 13125, 13545, 13965, 14025, 14175, 14355, 14595, 14805, 15015, 15435, 15645, 15675, 16005, 16065, 16275, 16335, 16695, 16905, 17325, 17745, 17955, 18135, 18375, 18585, 18795, 19215, 19635, 20475, 20685, 21105, 21315, 21525, 21945, 22365, 22605, 22995, 23205, 23595, 23625, 23835, 24255, 24675, 24885, 24915, 25245, 25515, 25725, 25935, 26325, 26565, 26775, 27027, 27195, 27885, 28035, 28215, 28245, 28275, 28665, 28875, 29295, 29925, 30195, 30345, 30555, 30723, 30765, 31185, 31365, 31395, 31605, 31815, 32025, 32175, 32235, 32445, 32835, 33075, 33345, 33495, 33915, 34125, 34155, 34485, 34515, 34755, 34965, 35175, 35805, 36225, 36435, 36645, 36795, 36855, 37275, 37485, 38115, 38745, 38955, 39165, 39195, 39375, 39435, 39585, 39765, 39795

There are 238 totients and 267 sigmas with 154 in common (permalink).

Totient Function: Jagged Versus Rounded Local Minima

If we plot the totients of the natural numbers against the values of the numbers themselves then we typically find an UP-DOWN or DOWN-UP alternation depending on where you start. Figure 1 shows such a plot for numbers in the range between 40 and 60.

Figure 1

Occasionally however, we see a different pattern. Figure 2 shows the totients of numbers plotted against the numbers themselves in the range between 300 and 330.

Figure 2: permalink

Looking at the graph in Figure 2 we see that from 313 to 317 we have an DOWN-DOWN-UP-UP pattern, clearly visible as a rounded rather than a local UP-DOWN-UP minimum. The value of the local minimum is the totient of 315. So where else do these rounded local minima occur. We are looking for numbers \(n\) such that:$$ \phi(n-2)>\phi(n-1)>\phi(n) < \phi(n+1)<\phi(n+2)$$where \( \phi \) represents the totient function. In the case of 315 we have:$$ \begin{align} &\phi(313)>\phi(314)>\phi(315) < \phi(316)<\phi(317) \\ &312 >156>144<156<316 \end{align}$$where it can be seen that 313 and 317 are prime numbers. So where else do these local rounded minima occur in the range from 3 up to 40000? It turns out that there are 238 such minima with 315 being the first (permalink).

315, 525, 735, 1155, 1365, 1575, 1755, 1785, 1815, 1995, 2145, 2415, 2475, 2805, 3045, 3315, 3465, 3885, 4095, 4125, 4305, 4515, 4725, 4935, 5115, 5145, 5355, 5775, 6045, 6195, 6405, 6435, 6615, 6825, 7035, 7095, 7245, 7395, 7455, 7605, 7665, 8085, 8265, 8505, 8715, 8745, 8925, 9135, 9345, 9405, 9555, 9735, 9765, 9975, 10185, 10395, 10455, 10545, 10815, 10965, 11055, 11235, 11385, 11445, 11655, 11865, 12075, 12285, 12495, 12675, 12705, 12915, 13125, 13335, 13545, 13695, 13965, 14025, 14175, 14355, 14385, 14595, 14805, 14835, 15015, 15045, 15225, 15405, 15435, 15645, 15675, 15855, 16005, 16065, 16275, 16335, 16485, 16695, 16905, 17085, 17325, 17355, 17745, 17955, 18135, 18165, 18375, 18585, 18645, 18795, 18975, 19215, 19425, 19635, 19665, 20055, 20265, 20295, 20475, 20625, 20685, 20865, 20895, 21105, 21255, 21315, 21525, 21945, 22365, 22425, 22575, 22605, 22785, 22995, 23205, 23265, 23415, 23595, 23625, 23655, 23835, 23985, 24225, 24255, 24675, 24885, 24915, 25095, 25245, 25305, 25515, 25575, 25725, 25905, 25935, 26145, 26325, 26565, 26775, 26985, 27027, 27195, 27615, 27825, 27885, 28035, 28215, 28245, 28275, 28455, 28665, 28815, 28875, 29055, 29295, 29505, 29865, 29925, 30195, 30345, 30555, 30723, 30765, 30975, 31185, 31365, 31395, 31605, 31815, 32025, 32175, 32235, 32445, 32655, 32835, 32895, 33033, 33075, 33345, 33495, 33705, 33735, 33915, 34125, 34155, 34335, 34485, 34515, 34545, 34755, 34965, 35175, 35385, 35805, 36225, 36435, 36465, 36645, 36795, 36855, 37065, 37275, 37455, 37485, 37695, 37905, 38115, 38535, 38745, 38775, 38955, 39165, 39195, 39375, 39435, 39585, 39765, 39795

These numbers constitute OEIS A076773:

A076773   2-nadirs of phi: numbers k such that phi(k-2) > phi(k-1) > phi(k) < phi(k+1) < phi(k+2).

There are no rounded local maxima in the range up to one million.

Do we find these types of rounded local minima for other functions like the sigma function? 

We do indeed, although in the case of the sigma function there are only two numbers in the range up to 40000 and they are 17254 and 27754 (the first two members of the sequence OEIS A076774: permalink). 

However, rounded local maxima are far more common in the sigma function. In the range up to 40000, there are 267 numbers and they are (permalink):

315, 405, 525, 693, 765, 945, 1125, 1155, 1395, 1575, 1755, 1785, 1845, 1995, 2205, 2475, 2565, 2805, 2835, 3003, 3045, 3285, 3315, 3465, 3645, 3675, 3885, 4095, 4125, 4275, 4347, 4455, 4515, 4725, 4995, 5115, 5355, 5445, 5733, 5775, 5805, 6045, 6195, 6237, 6405, 6435, 6615, 6825, 6885, 7035, 7155, 7245, 7605, 7875, 7995, 8085, 8325, 8415, 8505, 8715, 8775, 8925, 9075, 9135, 9315, 9405, 9555, 9675, 9765, 9975, 10125, 10395, 10773, 11205, 11235, 11385, 11445, 11475, 11655, 12045, 12075, 12285, 12555, 12675, 12705, 12915, 13005, 13125, 13275, 13365, 13545, 13725, 13923, 13965, 14025, 14175, 14355, 14595, 14685, 14805, 15015, 15075, 15435, 15525, 15645, 15675, 15795, 16005, 16065, 16245, 16275, 16335, 16443, 16695, 16875, 16905, 16965, 17325, 17595, 17685, 17745, 17955, 18135, 18315, 18375, 18585, 18765, 18795, 19005, 19035, 19215, 19305, 19575, 19635, 19845, 20475, 20685, 20925, 21105, 21285, 21315, 21483, 21525, 21675, 21945, 22005, 22155, 22275, 22365, 22605, 22869, 22995, 23085, 23205, 23355, 23595, 23625, 23835, 23925, 24045, 24255, 24453, 24525, 24675, 24885, 24915, 25155, 25245, 25515, 25725, 25875, 25935, 26325, 26505, 26565, 26775, 27027, 27195, 27225, 27405, 27531, 27885, 28035, 28125, 28215, 28245, 28275, 28413, 28485, 28665, 28875, 28917, 29295, 29325, 29475, 29835, 29925, 30135, 30195, 30345, 30555, 30723, 30765, 31005, 31185, 31365, 31395, 31515, 31605, 31815, 32025, 32175, 32235, 32445, 32805, 32835, 33075, 33165, 33285, 33345, 33495, 33915, 34125, 34155, 34485, 34515, 34755, 34875, 34965, 35175, 35235, 35325, 35595, 35805, 35955, 36045, 36075, 36135, 36225, 36435, 36645, 36675, 36795, 36855, 37107, 37125, 37275, 37395, 37485, 37845, 38115, 38205, 38475, 38493, 38745, 38955, 39015, 39123, 39165, 39195, 39375, 39435, 39525, 39585, 39627, 39765, 39795

Let's take 405 as an example. It's position at the apex of a rounded local maximum is shown in Figure 3.

Figure 3: permalink

The above numbers constitute OEIS A323380:

A323380   Odd n such that sigma(n) > sigma(n+1) and sigma(n) > sigma(n-1), sigma = A000203.

Overall, these variations on the UP-DOWN or DOWN-UP alternation are an interesting topic to investigate and one that I'd not considered before.