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

[Antti Karttunen]

On Wed, Dec 31, 2014 at 2:02 PM, Antti Karttunen
<antti.karttunen at gmail.com> wrote:
> 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.

Or maybe my muddy half-forgotten intuition from a month ago was  based
on what I say in:

https://oeis.org/A250477 "Number of times p_n (the n-th prime) occurs
as the least prime factor among numbers 1 .. (p_n)^2 * p_{n+1}: a(n) =
A078898(A251720(n)). "

"a(n) = Position of 6 on row n of array A249821. This is always larger
than A250474(n), the position of 4 on row n, as 4 is guaranteed to be
the first composite term on each row of A249821.
Naturally also: p_n occurs more times as the smallest prime factor in
range [1, (p_n)^2 * p_{n+1}] than in range [1, (p_n)^3]."

And clearly that is quite self-evident.

Now, the growing initial prefix of fixed terms on successive rows of
https://oeis.org/A251722
correspond to the growing (<-- well, so far an assumption we want to
actually prove here) number of semiprimes in
https://oeis.org/A083221
between the square of prime at the position 2 of each row n, and the
cube of the same prime at the position A250474(n) of the same row n.

So any non-fixed terms on row n of https://oeis.org/A251722 must occur
at places where there is some number with A001222(n) >= 3 in the same
(A001222 = bigomega) position at array A083221 ?

And clearly it's always the cube of that prime(n) which is the first
number neither prime or semiprime on the row n of A083221.

Are we anything nearer?


Antti

>
> 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