Problems & Puzzles: Collection 20th Coll.20th-005. Questions about the function sigma(x)=n.

On March 12, 2018, Fred Schneider sent the following puzzle:

From this sequence, https://oeis.org/A002191, it can be seen that not all n are solutions to sigma(x) = n for some x where sigma is the sum of divisors.

Note that this sequence contains consecutive odd numbers 13 (=sigma (9)) and 15 (=sigma (8)).

Q1) Are there any other consecutive odd numbers that may be solutions to the sigma function?

Note also that this sequence actually contains solutions for 12 and 14 giving 4 consecutive n from 12-15 such that sigma(x) = n

Q2) Follow up to question #1, are there any chains are 3 or more consecutive numbers that have a solution (which include 2 or more odd numbers).

Q3) What is the longest chain of consecutive even numbers that you can find?

Regarding the 3rd question, the best streak I have found is 5. It first occurs at 612. I don't see anything larger than that through 10 million.  It's interesting because 5 seems to occur quite often.

Contributions came from Emmanuel Vantieghem, Giovanni Resta.

***

Emmanuel wrote:

Q1.
There are three other pairs below 100000000 :
91 = sigma(36), 93 = sigma(50)
241573 = sigma(241081), 241575 = sigma(117128)
38152387 = sigma(15069924), 38152389 = sigma(23011209)

Q2.
A chain of four consecutive numbers among the sigma images occurs only once below 10^8.
There are 518 chains ol lenght 3 (the chains  12, 13, 14  and  13, 14, 15  are included).

Q3.
A chain of  5  even consecutive sigma values occurs earlier : 36, 38, 40, 42, 44
(sigma values of  22, 37, 27, 20 (or 26 or 41), 43
There are longer chains :
the earliest of length 6 starting at  2896234 = sigma(2480233)
the earliest of length 7 starting at  13248726 = sigma(8832482)
the earliest of length 8 starting at  32474790 = sigma(21649858)

***

Giovanni wrote:

For what concerns consecutive odd numbers, my answer is
in sequence A300779: Odd numbers x such that x and x + 2 are both sums of divisors.

The known ones are (I found the last ones):
1, 13, 91, 241573, 38152387, 139415801707, 55342019130181, 61166380109329, 417542026135897, 417542026135897, 13805828672331787.

Note that one (417542026135897)
is repeated because it can be generated by two numbers (the pairs are in A300780).

Since they are so "spaced out" finding 3 consecutive odd numbers seems almost
impossible.

Apart 13, none of these numbers + 1 is a sigma value, so they
do not belong to triples or quadruples of consecutive values like for 12,13,14,15.

I'm working on the even consecutive numbers. Looking up to 10^11 will
not be a problem.

Looking at the first terms of sequence A002191:
1, 3, 4, 6, 7, 8, 12, 13, 14, 15, 18, 20, 24, 28, 30, 31, 32,
36, 38, 39, 40, 42, 44, 48, 54,...

it seems to me that 36,38,40,42,44 are 5 consecutive even numbers.
Since you say the first 5 consecutive even numbers start at 612 I wonder
if you overlooked it or if for some reason you discarted this set,
maybe because in between there is also an odd number (39).

...

For even values, I found the following runs, searching up to 10^12.

For every run, I report a set of numbers whose sigmas give the
values in the run.

Length Start  Witnesses
5           36  (22, 37, 27, 41, 43)

6      2896234  (2480233, 1930822, 2726137, 2878417, 2896241, 2005083)

7     13248726  (8832482, 13248727, 11354329, 13248731, 13248733,
13238557, 11389217)

8     32474790  (21649858, 13917756, 27834793, 32372357, 32474797,
31620163, 32474801, 32474803)

9   2585306188  (2584719733, 2585306189, 1107988356, 2216106193,
2579179457, 2585306197, 2583064747, 2585306201, 2585306203)

10  39528356220  (37332336413, 39528356221, 34587311689, 26352237482,
39528356227, 33883461829, 26340044342, 39528356233, 39527897317,
23974591762)

11 795029717004  (795029717003, 454302695428, 792230316313,
424015849064, 795029717011, 471803926293, 526665255262, 795029717017,
738241880077, 530019811346, 242325537600)

*** Records   |  Conjectures  |  Problems  |  Puzzles  