[seqfan] Odd perfect numbers, and empirical evidence from "split sigma" conditions?

[Antti Karttunen]

Some months ago I got hooked about the existence/non-existence of odd
perfect numbers, even though I have later come to realize it
fundamentally as quite a boring problem. In any case, because the real
opns are so rare, I started generating sequences with somewhat laxer
criteria to get some feel of the problem.

First, consider HAKMEM item #23 from
http://www.inwap.com/pdp10/hbaker/hakmem/boolean.html#item23

ITEM 23 (Schroeppel): (A AND B) + (A OR B) = A + B = (A XOR B) + 2 (A AND B).

The right hand side (or its optimized versions) is the reason your
computer can add two binary numbers together, while the left side
inspired me to submit the following four sequences last August:

A318456 a(n) = n OR (sigma(n)-n).
A318458 a(n) = n AND (sigma(n)-n).
A318466 a(n) = 2*n OR sigma(n).
A318468 a(n) = 2*n AND sigma(n).

So, for example, A318456(n) + A318458(n) = (n OR (sigma(n)-n))+(n AND
(sigma(n)-n)) = n + (sigma(n)-n)) = sigma(n).

Likewise, A318466(n) + A318468(n) = (2*n OR sigma(n)) + (2*n AND
sigma(n)) = 2n + sigma(n).

So, if n is a perfect number (whether even or odd), then we have:

2*A318456(n) = 2n = 2*A318458(n)

and

A318466(n) = 2*n = A318468(n).

Moreover, it follows that ALL the following 6 equivalences should hold:

2*A318456(n) = 2*A318458(n), 2*A318456(n) = A318466(n), 2*A318456(n) =
A318468(n),
2*A318458(n) = A318466(n), 2*A318458(n)  = A318468(n), A318466(n) = A318468(n),

and if ANY one of them does not hold on any particular n, that n
cannot be a perfect number.

So I submitted a sequences like https://oeis.org/A324647
"Odd numbers k such that 2k is equal to A318468(k) (bitwise-AND of 2*k
and sigma(k))."

What is apparent in terms of A324647, is that almost all of them
(apart from a few exceptions Giovanni Resta found later) have a prime
factorization that accords well with the condition Euler gave for odd
perfect numbers, that is, the sequence https://oeis.org/A228058 "Odd
numbers of the form p^(1+4k) * r^2, where p is prime of the form 1+4m,
r > 1, and gcd(p,r) = 1."

So far so good. However, later I came a different idea. Consider
A000203(n), also known as sigma, the sum of divisors of n.

Let's cut (or split) it in two parts (called x and y), with x + y =
sigma(n), with x and y both integers in Z.

This can be done based on some "natural phenomena", like that x is the
sum of certain kind of divisors of n, and y is the sum of the
complementary kind of divisors, or wholly arbitrarily.

Now we can subtract the other from n and n from the other, to get
(n-x) and (y-n).
Subtracting these in turn from each other gives (n-x) - (y-n) = 2n -
(x+y) = 2n - sigma(n) = A033879(n), the deficiency of n.

Thus, for deficiency to be zero (and thus n to be perfect), we must
have (n-x) = (y-n). And thus it must be also that (n-x) divides (y-n)
(or be zero) and vice versa.  An easy way to check these conditions is
to use gcd because it won't choke on zeros. E.g., we look for such n
that gcd(n-x,y-n) = abs(n-x) or likewise that it is equal to abs(y-n).

One natural way to split sigma(n) is to consider the sum of all of
its possible divisors (A000203), and sum of all its impossible
divisors (A000004). Then we look for n's satisfying gcd(n-0,
A000203(n)-n) = abs(n-0), i.e., n's satisfying gcd(n, sigma(n)) = n,
that is, the multiply perfect numbers A007691, that includes also all
perfect numbers.

Now, what is perhaps strange, in many cases using these kinds of
conditions seems to give us very different kinds of terms than what
Euler's criteria allows. We can also add additional constraints, like
that the quotient is positive and/or odd (like +1 is), or discard any
obvious false positives.

Like in A326064: "Odd composite numbers n, not squares of primes, such
that (A001065(n) - A032742(n)) divides (n - A032742(n)), where A032742
gives the largest proper divisor, and A001065 is the sum of proper
divisors" which is a bit of border-line case, with some of those
non-unitary prime factors being of the form 4k+1 and some of the form
4k+3.

While in A326074 "Numbers n such that 1+(A001065(n)-A020639(n)) is not
zero and divides 1+n-A020639(n)" the solutions seem to overwhelmingly
be nonsquare semiprimes.

However, the clearest signal so far is in
https://oeis.org/A325981 "Odd composites for which gcd(A325977(n),
A325978(n)) is equal to abs(A325977(n))."

Here the "cut" is based on a semi-artificial way of splitting sigma(n)
in two parts:
the average of {sum of unitary divisors} and {sum of squarefree
divisors} (A325973),
and its "complementary part", A325974, "average of {sum of non-unitary
divisors} and {sum of nonsquarefree divisors}".

Now, thanks to Giovanni Resta extending the b-file, we know that the
first 147 terms of A325981 are all of the form {one squared prime
factor and the rest of them unitary}, i.e., are terms of A072357,
where each term satisfies A001222(x) - A001221(x) = 1.

Now... you might say at this point, "Whooaa, what a fool, whenever a
real OPN arrives, it doesn't care at all how sigma(n) has been cut
into two parts, because the perfect numbers trump any such
constraints!".

But... I reply: "Nobody has proved that such odd monsters exist at
all, so let's see first what the empirical evidence tells us."

So, if we manage to prove that say, A325981 cannot afford any other
terms than those of A072357, not _even when_ the quotient
A325978(n)/A325977(n) would be exactly +1 (as it would be for opns),
then we would be done. That is, deduce it somehow _without_
considering what the whole sum of divisors would be, because then we
always come back to the square one.

Also, you may wonder why I concentrate on A325981, essentially the
quotient A325978(n)/A325977(n), instead of A325979, the quotient
A325977(n)/A325978(n), as both should be +1 for opn's. But A325981
seems much more regular in its prime factorization structure than
A325979, and the drunks like to search their car keys under the street
lamp. But of course, even in A325981 a first few thousand initial terms might
not tell the truth about its real character.

On the other hand, the mentioned A072357 is itself a subsequence of A048107,
"Number of unitary divisors of n (A034444) > number of non-unitary
divisors of n (A048105)." which could be a clue that there might
indeed be a real reason why A325981 cannot afford other kinds of
terms. Note also that "number of (non-)unitary divisors" = "number of
(non-)squarefree divisors".

At this point I would like to hear an expert opinion about the
utility/futility of creating such split-sigma gcd (divisibility)
conditions? Is such "empirical evidence" of any real value here?

For what is worth, I initiated a new index-entry
"Index entries for sequences where any odd perfect numbers must occur"
and you can find sequences belonging to that category by searching
"link:opnseqs" as with:
https://oeis.org/search?q=link%3Aopnseqs&sort=&language=&go=Search

Of course, to prove the nonexistence of opns, it suffices to pick a
set of two or
more such sequences and show that it is impossible for any number to
occur in all of them at the same time. So maybe it's good to have some
variety there.
(But how many of the conditions are really independent of each other?)

See also https://oeis.org/A019283 and https://oeis.org/A326181 for a
more traditional approach involving triply perfect numbers. There must
be a lot more of sequences that fit to the subcategory "sequences that
satisfy some other condition". You may add them there if something
especially pertinent comes up. (However, not necessarily sequences
like A005408).


Best regards,

Antti