Problems & Puzzles:
Joseph L. Pe proposes the following puzzle:
I would like to propose the following problem whose answer I
don't know. (It might be easy.) Let sigmaO(n) = the sum of the
odd (positive) divisors of n. The odd abundance of n is defined
similarly to abundance as: oa(n) = sigmaO(n) / n.
Is there a (positive) integer N with N > 1 whose odd abundance
oa(N) is an integer?
I have checked all values of n up to 10^6, but for these values,
oa(n) is never an integer.Btw, the problem is very easy if we
consider the even or the ordinary abundances.
Contributions came from Faride Firoozbakht, Fred Schneider, Luke Pebody
& Dan Dima.
Faride wrote: "It seems that there is no such number. But the proof of
this conjecture is likely impossible."
Say m> 1 is an odd number:
Since 2 does not contribute to the sum of divisors of sigmaO, for any
k>= 0 sigmaO(m*2^k)= sigmaO(m)
So, if want to find sigmaO(n)/n to be an integer, it makes sense that we
search only for odd numbers (for any solution n where n= m*2^k, m would
also have to be a solution).
Clearly, sigmaO(n)=sigma(n) for odd n.
To find an odd n such that, for sigma(n)/n = 2, we would have to find
an odd perfect number (of which none are known to exist).
In fact for any integers q>=3, there are no solutions for sigma(n)/n=q.
In other words, there are no known odd multiply perfect numbers.
Puzzle 332 is equivalent to the following question: "Is there an odd
multiply perfect number?" It is a longstanding conjecture that the
answer is no.
If n is the largest odd factor of N, then oa(n)=oa(N). Thus if oa(N)/N
is an integer, then oa(n)/n=(N/n)*oa(N)/N is an integer. This is the sum
of the factors of n, divided by n.
Suppose there is a positive integer N with N > 1 whose odd abundance
sigmaO(N) / N is an integer. Let N = 2^k * M, where M - odd positive
integer. Then: sigmaO(N) = sigmaO(M) = sigma(M) Hence: sigma(M) / M =
2^k * (sigmaO(N) / N) is an integer too, which means that M must be a
multiperfect number whose sum of divisors is divided by the number
I found this page very interesting too:
An old and unproved conjecture states that the only odd multiply perfect
number is 1 - which extends the ancient Greek's conjecture that states
the only odd perfect number is 1. Obviously, if the conjecture holds
then there is no positive integer N with N > 1 whose odd abundance
sigmaO(N) / N is an integer. It seems very likely such integer N does
not exist. Any hypothetical solution to this puzzle will lead to the
discovery of an odd multiperfect number! Conversely this is not true
since an hypothetical odd multiperfect number may not lead to a
solution!. Richard Schroeppel constructed all MPNs < 10^70, and proved
that there exist exactly 258 MPNs smaller than 10^70 and obviously no
odd MPNs smaller than 10^70.