[seqfan] Re: Help needed... for proofs, Brocard's Conjecture, etc.

[Antti Karttunen]

On Tue, Dec 30, 2014 at 10:22 PM, Antti Karttunen
<antti.karttunen at gmail.com> wrote:
> Here are two things for which I need help right now (not just the end
> of the next year):
>
> In https://oeis.org/A050216 "Number of primes between (prime(n))^2 and
> (prime(n+1))^2, with a(0) = 2 by convention." it is told that
> Brocard's Conjecture states that for n >= 2, a(n) >= 4.
>
> See also http://en.wikipedia.org/wiki/Brocard%27s_conjecture
>
> Now the question: for up to which k it is proved (or "obvious") that
> A050216(n) >= k, for all n >= 2 ?
>
> For example, is it clear that https://oeis.org/A251723 "First
> differences of A054272, A250473 and A250474: a(n) = A054272(n+1) -
> A054272(n). " (and also "One less than A050216") is always positive (>
> 0) or even nonnegative (>= 0) ?
> I.e. that a sequence like https://oeis.org/A054272 is (genuinely) growing?

Note that if what I say at
https://oeis.org/A251722

"On row n the first non-fixed term is A250474(n+1) at position
A250474(n), i.e., on row 1 it is 5 at n=4, on row 2 it is 9 at n=5, on
row 3 it is 14 at n=9, etc. All the previous A250473(n) terms are
fixed."

is really true in its entirety, then the latter sentence "All the
previous A250473(n) terms are fixed." (i.e. that the A250474(n) is
really the _first_ non-fixed term) automatically implies that
https://oeis.org/A250474
must be genuinely growing (because otherwise we get a contradiction),
i.e. that A251723 (n) >= 1 for all n, and thus A050216(n) >= 2 for all
n. (So at least half of that 4 stipulated by Brocard's conjecture).

However, the problem is that I'm not anymore as sure of my claim's
truth as when I submitted those sequences.

Clearly, "On row n A250474(n+1) occurs at position A250474(n), i.e.,
on row 1 it is 5 at n=4, on row 2 it is 9 at n=5, on row 3 it is 14 at
n=9, etc." holds,
but whether it is indeed the _first_ non-fixed term, I'm not so
certain anymore. I had probably silently taken for granted that
A250474 is genuinely growing when coming to that conclusion, but I
realize now that is just the monotonicity of A250474 that begs for a
proof.

Furthermore, I'm not sure whether playing with any of these
permutations between arrays A083221 and A246278 really help at all, or
whether they just obfuscate this issue.


Cheers,

Antti


>
>
> ----------
>
> Secondly, can somebody prove at https://oeis.org/A251726
> "Numbers n > 1 for which there exists r <= gpf(n) such that r^k <=
> spf(n) and gpf(n) < r^(k+1) for some k >= 0, where spf and gpf
> (smallest and greatest prime factor of n) are given by A020639(n) and
> A006530(n). "
>
> my conjecture that:
> "If any n is in the sequence, then so is A003961(n)."
>
> where A003961 shifts the primes in the prime factorization of n one
> step towards larger primes,
> thus also spf(n) and gpf(n) will be replaced by the respective
> nextprimes. Note that as far as I see it, this is towards "unsafe
> direction", concerning the defining condition of A251726, because the
> "old r" doesn't necessarily work anymore, but a larger value is
> sometimes needed.
>
>
> In any case, I am myself absolutely lousy with any proofs involving
> limits, inequivalences or contradictions.
>
>
> Thanks in advance,
>
> Antti