[seqfan] Re: A344826

[M. F. Hasler]

On Tue, Jun 15, 2021 at 10:08 AM David Corneth <davidacorneth at gmail.com>
wrote:

> (...)

Maybe with some refining this method could speed the search up a little to
> at least get upperbounds of other terms or maybe even actual proven new
> terms.
>

It's not difficult to construct new terms through the method alluded to in
the COMMENTS,
which I've tried made a little more precise/explicit through a new comment
(pending edit):
If k = a(n) with q = A343886(k) = k/A097621(k), and m=A961(q) is relatively
prime to k,
then A097621(k*m) = A097621(k) * A097621(m) = A097621(k) * q
so  k*m / A097621(k*m) = q * m / q = m  is an integer and k*m is in the
sequence.

One might call "primitive" those terms that are not derived in this way
from smaller terms.
All the terms listed so far are primitive, but 5 of the last 9 terms,
a({35, 36, 38, 42, 43} = {694008, 1097712, 1778400, 2794176, 3470040}
can be "extended" in that way to 5 new (non primitive) terms
(7634088, 18661104, 12448800, 64266048, 38170440).
These have the  q-values (11, 17, 7, 23, 11), so the second and 4th again
produce new terms 541172016 and 3020504256 with q-values 29 resp. 47, so
both produce new terms 36258525072, 413809083072 with q-values 67, 137,
producing 8665787492208, 258630676920000 with q-values 239, 625,
producing 11274189527362608, 1101508053002280000 with q-values 1301, 4259,
producing 114647233303750360752, 43759610421621577560000 with q-values
10169, 39727,
which are again prime, and so it goes on.
The probability that m = A961 <https://oeis.org/A000961>(q) is a prime (and
then larger than all previous prime factors)
is growing as q grows, and in that case we get a new terms.
This is a sufficient, but not necessary condition: e.g., 625 in the above
is not prime
[but is coprime to 413...72 and therefore produces a new term],
but m = A961 <https://oeis.org/A000961>(625) = 4259 is again prime and
therefore the chain continues.

So the main difficulty is, as often, not to have "holes" among known terms.

And the main problem is that it is entirely based on oeis.org/A097621 which
is actually **plain wrong**!
Indeed, it is defined as
A097621 : In canonical prime factorization of n replace the k-th prime
power with k.But, instead of
A246655 Prime powers: numbers of the form p^k where p is a prime and k >= 1.
it uses
A000961 Powers of primes. Alternatively, 1 and the prime powers (p^k, p
prime, k >= 1).
If it did use  A246655 as intended in the definition,
then A097621 would be not just slightly, but completely different,
and as a consequence, even more totally different would be this sequence
oeis.org/A344826 "Integers k such that k/A097621(k) is an integer."
This proves that A097621 and A344826 don't really have a mathematical
meaning,
they are dependent on the indices arbitrarily assigned to the prime powers.

So the above construction and reasoning may be "interesting" in some sense,
and they would still be valid, but would lead to completely different
sequences,
if the values of A097621 would correspond to its current definition.
(One might "fix" that definition but obviously it does not make any sense
to use A961 including 1 in front of the prime powers, when one is only
considering the true prime powers (> 1) which appear in factorisations.

- Maximilian

On Mon, Jun 14, 2021 at 6:13 PM Tom Duff <eigenvectors at gmail.com> wrote:
>
> > I think my question wasn't clear enough.
> > Why is this a mathematically interesting sequence?
> > Is the sequence related to some mathematically interesting problem?
> > Does it shed light on some important question?
> > The number of terms in the b file doesn't address those questions.
> >
> > I'm unlikely to be interested in undertaking a long computation if it
> > doesn't provide some mathematical insight.
> >
> > On Sat, Jun 12, 2021 at 1:44 PM <michel.marcus at free.fr> wrote:
> > >
> > > Well, I see several reasons:
> > > 90 terms for A344826 and A343886 is not much; 2*10^8 is not that big
> too.
> > > they are similar to A127724 and so somehow to A127724 (but of course,
> > with a different function).
> > > my pari script is limited to the allocatemem I can do, maybe someone
> has
> > better.
> > > I tried to save the underlying data in files to counteract my memory
> > limitation but when n increase, it kept switching files, so it did not
> work.