Regulars = Divisors + Semidivisors
http://global.britannica.com/EBchecked/topic/496213/regular-number
So for example:
- 6 has 5 regulars: 1, 2, 3, 4, 6.
- 8 has 4 regulars: 1, 2, 4, 8.
- 9 has 3 regulars: 1, 3, 9.
- 10 has 6 regulars: 1, 2, 4, 5, 8, 10.
- 12 has 8 regulars: 1, 2, 3, 4, 6, 8, 9, 12.
- A prime number has 2 regulars: 1 and itself.
I've already found the quantity of regulars of every primorial less than 223092870:
- 2 has 2 regulars
- 6 has 5 regulars
- 30 has 18 regulars
- 210 has 68 regulars
- 2310 has 283 regulars
- 30030 has 1161 regulars
- 510510 has 4843 regulars
- 9699690 has 19985 regulars
(Maybe there are a few mistakes, so if someone could also check the 4 last ones just to be sure it would be very nice of you.)
Once I get the quantity of regulars of 223092870 and 6469693230 I will finally be able to discover the secret of primorials (or maybe not). And I shall share this secret with the person who finds me the quantity of regulars of these two primorials =)
23# : 83074
29# : 349670
31# : 1456458
37# : 6107257
41# : 25547835
43# : 106115655
Edit: I don't have the required 50 rep to comment, so I have to do comment on gammatester's answer from here: Did you use 32-bit integers to bruteforce it? That could have lead to negative numbers and more solutions than there really are. Also your $k_i$'s shouldn't be bounded by $\alpha_i$, should they?