I'm analyzing integer numbers with factor pair sums forming an arithmetic sequence. E.g., the number 27720 allows to build the arithmetic sequence 334..338..342..346 :
27720 = 154 x 180 and 154+180=334
27720 = 140 x 198 and 140+198=338
27720 = 132 x 210 and 132+210= 342
27720 = 126 x 220 and 126+220=346
It seems difficult to find integer numbers with longer sequences of this type. Thanks for any help.
This is going to be more of a response than an "answer" (I suspect if there were a well-understood solution for the implied problem, someone would have already posted it.) I see this as yet another example of how addition "doesn't play well together" with multiplication; if they did, a lot of number-theoretic conjectures would have been resolved long ago...
The integer in your example is $ \ 27720 \ = \ 2^3·3^2·5·7·11 \ \ , \ $ for which $ \ \sqrt{27720} \ \approx \ 166.5 \ \ . \ $ If we write out its divisors, "folding back on themselves" those which are closest to the square-root of the number, we have
Each of the pairs of divisors you found are marked in a particular color. They "partition" the factors as:
$$ 154 · 180 \ \ = \ \ (2·7·11) \ · \ (2^2·3^2·5) \ \ \ , \ \ \ 140 · 198 \ \ = \ \ (2^2·5·7) \ · \ (2 ·3^2·11) \ \ \ , $$ $$ 132 · 210 \ \ = \ \ (2^2·3·11) \ · \ (2·3·5·7) \ \ \ , \ \ \ 126 · 220 \ \ = \ \ (2·3^2·7) \ · \ (2^2 ·5·11) \ \ \ . $$
A search among divisors "near" the square-root of the integer seems more fruitful, since the divisors greater that $ \ \sqrt{N} \ $ "spread out" more and more "rapidly", making it less likely that sums of pairs of divisors will be uniformly-spaced, as we desire for arithmetic progressions. (I won't say such progressions cannot occur -- I have not studied that issue exhaustively.)
So it appears that one wants to look at integers with lots of divisors for the best prospects of finding such progressions. If we examine $ \ 2751840 \ = \ 2^5·3^3·5·7^2·13 \ \ , \ \ \sqrt{2751840} \ \approx \ 1658.9 \ \ , \ $ we find "in the vicinity" of $ \ 1659 \ \ : $
The arithmetic progression I located is $$ \ 1638 + 1680 \ = \ 3318 \ \ , \ \ 1512 + 1820 \ = \ 3332 \ \ , \ \ 1456 + 1890 \ = \ 3346 \ \ . \ $$ (There may be others, but none stood out to me while checking differences between divisor-pair sums.)
Testing an integer with a still larger number of prime factors - - $ \ 5054400 \ = \ 2^6·3^5·5^2·13 \ \ , $ $ \sqrt{5054400} \ \approx \ 2248.2 \ \ $ - - the divisors around the square-root are
In this case, I belatedly found an arithmetic progression, though not close to the square-root:
$$ \ 1350 + 3744 \ = \ 5094 \ \ , \ \ 1296 + 3900 \ = \ 5196 \ \ , \ \ 1248 + 4050 \ = \ 5298 \ \ . \ $$ [This was "edited into" the orginal illustration.] It certainly does not appear to be easy to find arithmetic progressions of length greater than $ \ 3 \ . $ This example also makes clear that it isn't necessarily enough for an integer merely to have a lot of divisors.
What we are seeking to do is locate pairs of divisors of an integer $ \ N \ $ for which we have $$ r + \frac{N}{r} \ = \ S \ \ , \ \ (r - \Delta_1) + \frac{N}{(r - \Delta_1)} \ = \ S + d \ \ , \ \ (r - \Delta_2) + \frac{N}{(r - \Delta_2)} \ = \ S + 2d \ \ , \ \ \ldots $$
The difficulty in locating these progressions seems to lie in the fact that there is no simple relation among the $ \ \Delta_k \ \ $ that readily tells us how to determine the initial term of the progression, the common difference between its terms, or what divisor-pair we should "start" at in what integer.
It struck me as I was putting this post together that it might be worth looking at the highly composite integers , each of which has more divisors than any of the integers which precede it ( $ \ 27720 \ $ is on the list). For $ \ 665280 \ = \ 2^6·3^3·5·7·11 \ \ , \ \ \sqrt{665280} \ \approx \ 815.6 \ \ , \ $ we note
Here, we still don't see any arithmetic progressions of length greater than $ \ 3 \ \ , \ $ but we do have ones which are consecutive and even interleaved!
$$ \ 792 + 840 \ = \ 1632 \ \ , \ \ 770 + 864 \ = \ 1634 \ \ , \ \ 756 + 880 \ = \ 1636 \ \ ; $$ $$ \ 720 + 924 \ = \ 1644 \ \ , \ \ 693 + 960 \ = \ 1653 \ \ , \ \ 672 + 990 \ = \ 1662 \ \ ; $$ $$ \ 756 + 880 \ = \ 1636 \ \ , \ \ 704 + 945 \ = \ 1649 \ \ , \ \ 672 + 990 \ = \ 1662 \ \ . \ $$ $$ \ 594 + 1120 \ = \ 1714 \ \ , \ \ 576 + 1155 \ = \ 1731 \ \ , \ \ 560 + 1188 \ = \ 1748 \ \ . \ $$
All this doesn't answer your question, but perhaps gives some sense of the depth of the problem. I would certainly be interested in remarks by other people here.