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).