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

[M. F. Hasler]

The comment in https://oeis.org/A050216,

In the n-th step of the sieve of Eratosthenes, all multiples of
prime(n) are removed. Then a(n) gives the number of new primes
obtained after the n-th step.

is at least not very precise. One cannot speak of "new primes
obtained", only for example about the new "minimal" upper bound, up to
which one is sure to have *eliminated* all composites. Yet, this is
not precise, either...

MH

On Wed, Dec 31, 2014 at 9:03 AM, Antti Karttunen
<antti.karttunen at gmail.com> wrote:
> 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
>
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