[seqfan] Re: https://oeis.org/A254211 Finite?

[M. F. Hasler]

On Thu, Jan 29, 2015 at 7:33 PM, Neil Sloane <njasloane at gmail.com> wrote:
> MFH's sequence A254337 is very nice.

Thanks ! Actually, I find the sister sequence A254341 (where every
other term is required to be odd) almost a bit nicer, because of this
"even monotony".

> Could someone please create a b-file, as large as possible,
> of course WITHOUT assuming the conjecture is true (that every composite
> even number appears and no odd number > 1 appears)?
>
> Is there a simple proof that 9 does not appear?

I don't know whether we find a rigorous proof, but the farther one
goes, the less it is probable that an odd term appears.
In some sense, the odd terms "could well appear" (cf A254341), but it
would be a little bit larger than the next possible even term
(essentially because there are so many odd numbers that are prime :
after 9 there is only 15 below 20), and so they get "further and
further pushed away".
There is only obstruction to have a(3)=9, namely a(2)=8  yields
a(2)+a(3)=17, prime.
Because  8+15=23 and 8+21=29,  the smallest odd integer that could
appear here is a'(3)=25, but of course a(3)=6 (and 10,12,...,24) are
smaller.
Then what odd composite could appear after 1,10,6 ?
Again only one obstruction (8+6+9 = 23) against 9,
and one (8+6+15=29) against 15, now already a'(4) = 21 would be
possible (6+21=27 and 8+6+21=35 both composite) but it is larger than
a(4)=10.

To get a hunch, we can count the number of "obstructions" against
placing 9 after a(n), i.e. the number of primes of the form
sum(i=k...n,a(i))+9.
We find:
0,0,1,1,1,2,1,1,3,1,2,3,8,6,6,6,8,9,4,3,5,6,7,6,8,4,5,10,6,10,7,8,13,11,6,10,7,17,12,13,11,9,12,9,7,19,12,13,18,6,10,12,20,11,10,19,
14,15,11,18,14,9,14,16,14,24,15,14,18,16,11,19,20,15,20,15,29,18,20,17,16,15,17,23,10,15,16,29,18,28,23,12,25,26,23,18,23,19,16,
It seems that this sequence (its lim inf) is increasing, although
there are some surprisingly low values at larger indices, e.g., 6 at
n=50, 9 at n=62.
I think these two are "lower record values" (i.e., all following terms
are larger ; other such records are b(10)=1, b(11)=2, b(20)=3,
b(26)=4, b(27)=5, b(50)=6, b(62)=9, b(85)=10, b(92)=12, b(147)=16,
b(181)=22, b(211)=25, b(239)=28, b(244)=30,...).

One could also study the smallest odd composite that could be placed
after a(n) :
9, 9, 25, 21, 39, 25, 69, 65, 45, 119, 95, 77, 55, 27, 595, 561, 531,
865, 1519, 1479, 1437, 1391, 1353, 1309, 1257, 1209, 1155, 1105, 1047,
2317, 2255, 2191, 3565, 5719, 13067, 12995, 12925, 12851, 12771,
12695, 12617, 12531, 12449, 12365, 12275, ...
We see that these also (more or less steadily) increase (because there
are more and more conditions on compositeness -- even though they do
not necessarily include the preceding ones)...

So it seems extremely improbable that 9 could ever occur, although I
can't wee how to prove this rigorously. (Maybe using a(n)~2n and good
estimates on prime gaps?)

Maximilian

>> > Am 27.01.2015 um 12:26 schrieb Ron Hardin:
>> >> Right, the every partial sum from either end is never prime.
>>
>> This inspired me https://oeis.org/draft/A254337
>>
>> Lexicographic first sequence such that a(k)+...+a(n) is never prime
>> and all terms are different.
>>
>> 0, 1, 8, 6, 10, 14, 12, 4, 20, 16, 24, 18, 22, 28, 26, 34, 30, 32, 36,
>> 40, 42, ...
>>
>> The sequence may not contain any prime. Does it contain all even
>> numbers > 2? (It seems so.) Does it contain any odd number beyond 1?
>> (It seems not.)