[seqfan] Re: A351913: numerator(1/d - 1/n) [was: Re: Upper bound for A091895 and A091896]
M. F. Hasler
oeis at hasler.fr
Wed Apr 13 15:30:59 CEST 2022
> On Tue, Apr 5, 2022 at 10:51 AM Michel Marcus wrote:
>
>> A bit different, did you see A351913 <http://oeis.org/A351913> ?
>> Any idea when can we say that Unknown can be set to -1 ?
>
>
This is an interesting question, I'd like to make it more explicit to
encourage people on this list who might be able to establish some relevant
results.
The problem is to show that there is no n such that numerator( 1/d(n) -
1/n ) = 102,
for example (first Unknown), where d(n) is the number of divisors of n.
Using d(p^e) = e+1 one can search solutions of the form n = 2^k m where
m is odd.
To get an even numerator x, one must have
(k+1) d(m) = 2^k (1+2E) for some integer E >= 0, which then leads to
m = 1+2E + 2^k g x, where g = gcd( m(1+2E), m-1-2E ).
Maybe one can scan the space of solutions by increasing values of k and E,
(k+1 must divide 2^k (2E+1), i.e., k = t*2^K - 1 with t | 2E+1 ;
K=0,1,2,...)
for each of which one can make the list of possible prime signatures of m,
(cf. oeis.org/A353248) and show that there is no solution m for any of
these.
But there are still some missing pieces to be filled in here, be it just
for x=102...
- Maximilian
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