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 in 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: $$\phi(313)>\phi(314)>\phi(315) < \phi(316)<\phi(317) \\ 312 >156>144<156<316$$ 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.