Geometric/arithmetic progression of divisor, quotient, remainder

[Nick Hobson]

Hi Seqfans,

Is this sequence worth adding?

6, 9, 12, 20, 28, 30, 34, 42, 56, 58, 65, 72, 75, 90, 110, 126, 132, 156,  
182, ... .

The sequence comprises all positive integers n such that there exist  
positive divisor d, quotient q, and remainder r in geometric progression.   
The order of terms in the geometric progression is left unspecified, but  
clearly must be one of q < r < d, r < q < d, or r < d < q.  For example,  
58 is in the sequence because 58 = 9*6 + 4.

Clearly the sequence is infinite.  There are 14, 65, 278 terms not  
exceeding 100, 1000, 10000, respectively.  I don't know its asymptotic  
density.

It's not difficult to show that the sequence misses the primes.   
Interestingly, at least initially, it also hits very few perfect powers,  
the only hits below 10^13 being 9, 10404, 16900, 97344, 576081, 6230016,  
7322436, 12006225, 36869184, 37344321, 70963776, 196112016, 256160025,  
1361388609, 1380568336, 8534988225, 9729849600, 12551169024, 13855173264,  
16394241600, 123383587600, 142965659664, 547674002500, 1812792960000,  
1882109610000, and 3602897496900, none of which are cubes or higher prime  
powers.

The corresponding sequence where there exist d, q, r in arithmetic  
progression; see below; seems to be denser.  There are 66, 782, 8349 terms  
not exceeding 100, 1000, 10000, respectively.  I don't find this sequence  
(or its complement) as interesting; for one, it doesn't appear to have any  
nice, simple properties, unless someone can find one...

2, 5, 7, 8, 11, 14, 16, 17, 18, 19, 20, 21, 23, 26, 29, 31, 32, 33, 34,  
35, 36, 38, 40, 41, 42, 44, 46, 47, ... .

Nick