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

[Antti Karttunen]

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?


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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