[seqfan] Re: conjectured largest number which is not the sum of an n-almost prime and a prime.

[Jonathan Post]

Zak,

Do feel free to submit that as seq and b-list of the lesser of the
seqs I've been exploring, with you as my co-author..

"Largest number which is not the sum of a prime and an n-almost prime"
is the harder one, which needs to be checked to 10^10 or so for n = 2,
3, 4, 5, 6 to have more confidence in the values scraped from existing
seqs.

On Sun, Mar 29, 2009 at 11:50 PM, zak seidov <zakseidov at yahoo.com> wrote:
>
>
> Dear Jonathan,
> here's list of first 1000 terms:
> {0, 0, 0, 0, 0, 1, 1, 1, 0, 2, 3, 2, 2, 1, 2, 2, 5, 1, 2, 2, 3, 2, 4, 2, 3,
> 3, 5, 5, 4, 1, 2, 4, 5, 2, 4, 3, 5, 6, 4, 5, 6, 3, 4, 5, 6, 5, 4, 3, 4, 4, 8,
> 7, 6, 4, 3, 7, 8, 6, 4, 4, 3, 10, 7, 6, 7, 4, 6, 10, 7, 6, 5, 6, 4, 7, 8, 9,
> 7, 5, 6, 9, 8, 9, 4, 5, 7, 8, 9, 11, 8, 4, 4, 11, 12, 10, 6, 10, 7, 13, 9, 9,
> 6, 7, 5, 10, 12, 10, 6, 10, 6, 10, 10, 9, 9, 8, 6, 11, 12, 13, 7, 5, 6, 16,
> 12, 14, 9, 9, 6, 15, 10, 11, 7, 11, 6, 13, 14, 15, 8, 13, 5, 13, 13, 10, 7,
> 11, 10, 15, 13, 17, 8, 8, 6, 19, 14, 11, 8, 11, 7, 17, 11, 15, 11, 15, 10,
> 17, 16, 18, 5, 10, 8, 13, 14, 17, 9, 15, 9, 14, 14, 14, 8, 11, 6, 19, 14, 14,
> 13, 13, 9, 21, 17, 18, 6, 16, 7, 17, 16, 18, 10, 14, 7, 19, 15, 18, 10, 16,
> 12, 20, 15, 20, 10, 6, 9, 20, 16, 20, 10, 20, 11, 20, 17, 21, 12, 22, 9, 22,
> 21, 21, 6, 21, 9, 21, 18, 24, 12, 16, 10, 18, 20, 19, 11, 14, 7, 23, 15, 22,
> 15, 18, 9, 25, 18, 24, 8, 18, 9, 20, 20, 23, 11, 22, 9, 23, 18, 22, 8, 19,
> 12, 26, 19, 22, 11, 17, 8, 27, 22, 19, 12, 22, 11, 26, 16, 21, 12, 21, 9, 26,
> 26, 21, 11, 21, 10, 27, 19, 26, 11, 21, 11, 28, 22, 21, 9, 19, 11, 25, 19,
> 22, 15, 27, 13, 30, 18, 25, 9, 25, 10, 28, 27, 27, 14, 30, 12, 29, 21, 28,
> 11, 22, 14, 34, 17, 27, 13, 16, 13, 34, 20, 28, 11, 25, 13, 27, 18, 28, 14,
> 29, 13, 25, 26, 30, 7, 29, 9, 30, 22, 31, 12, 22, 13, 30, 27, 29, 12, 25, 9,
> 34, 24, 26, 16, 30, 12, 34, 22, 29, 13, 30, 14, 29, 26, 28, 14, 26, 11, 27,
> 19, 32, 11, 33, 18, 27, 22, 35, 15, 18, 12, 31, 26, 32, 14, 30, 13, 39, 26,
> 35, 15, 36, 13, 31, 32, 31, 13, 32, 12, 31, 24, 33, 15, 32, 17, 33, 23, 35,
> 13, 19, 9, 34, 21, 35, 18, 32, 15, 35, 26, 33, 12, 35, 12, 32, 30, 34, 12,
> 33, 13, 37, 25, 31, 14, 36, 16, 32, 23, 36, 13, 30, 13, 34, 28, 38, 20, 31,
> 14, 41, 26, 36, 13, 26, 13, 35, 32, 37, 13, 35, 17, 37, 29, 33, 16, 37, 18,
> 38, 26, 41, 11, 30, 13, 39, 30, 34, 18, 37, 15, 42, 27, 37, 17, 40, 16, 39,
> 36, 38, 17, 39, 13, 43, 27, 33, 13, 36, 21, 32, 26, 38, 15, 29, 13, 43, 30,
> 37, 16, 43, 16, 46, 26, 41, 15, 39, 11, 40, 43, 36, 13, 43, 17, 46, 28, 39,
> 16, 46, 17, 34, 29, 37, 17, 38, 13, 45, 27, 42, 23, 35, 15, 43, 27, 41, 14,
> 46, 17, 44, 34, 45, 17, 44, 13, 44, 32, 40, 12, 46, 19, 40, 30, 41, 14, 37,
> 14, 46, 28, 42, 20, 44, 14, 44, 25, 39, 18, 42, 15, 45, 40, 42, 11, 40, 15,
> 44, 26, 36, 14, 45, 19, 41, 26, 42, 14, 40, 9, 48, 24, 43, 19, 42, 11, 41,
> 30, 41, 12, 41, 15, 35, 30, 40, 13, 45, 11, 42, 25, 36, 14, 38, 19, 42, 30,
> 33, 15, 29, 11, 46, 27, 42, 16, 45, 19, 39, 24, 41, 15, 44, 12, 41, 34, 41,
> 11, 43, 13, 38, 28, 42, 12, 41, 17, 36, 26, 35, 12, 32, 11, 38, 25, 41, 21,
> 39, 13, 43, 24, 37, 12, 40, 15, 39, 28, 40, 13, 44, 15, 37, 25, 40, 12, 44,
> 14, 43, 21, 39, 13, 35, 9, 46, 33, 35, 14, 44, 15, 39, 22, 42, 15, 42, 13,
> 36, 31, 33, 12, 55, 15, 46, 23, 35, 13, 39, 19, 38, 26, 39, 12, 41, 14, 41,
> 26, 46, 17, 44, 11, 48, 23, 43, 13, 36, 13, 38, 34, 35, 15, 45, 13, 44, 25,
> 37, 11, 47, 16, 34, 22, 38, 12, 42, 12, 39, 25, 39, 13, 43, 13, 44, 23, 46,
> 14, 47, 14, 38, 34, 40, 11, 43, 13, 42, 23, 33, 12, 45, 14, 38, 29, 41, 11,
> 38, 15, 42, 24, 42, 15, 46, 12, 37, 22, 44, 13, 43, 15, 37, 30, 46, 14, 40,
> 12, 42, 24, 40, 12, 45, 21, 38, 24, 42, 11, 45, 12, 41, 25, 39, 16, 41, 13,
> 37, 27, 46, 13, 48, 12, 39, 33, 45, 8, 44, 12, 41, 20, 36, 15, 46, 14, 38,
> 25, 38, 9, 35, 13, 38, 21, 43, 19, 46, 16, 41, 24, 41, 11, 46, 12, 39, 32,
> 37, 12, 39, 10, 48, 24, 41, 10, 45, 17, 35, 22, 42, 13, 40, 13, 38, 24, 41,
> 16, 39, 12, 38, 21, 43, 12, 43, 12, 36, 32, 39, 9, 35, 15, 43, 24, 35, 11,
> 49, 14, 38, 28, 36, 14, 39, 15, 47, 27, 40, 15, 42, 15, 35, 22, 40, 11, 50,
> 11, 36, 26, 44, 15, 46, 11, 39, 23, 34, 11, 41, 21, 40, 20, 39, 13, 44, 15,
> 40, 23, 35, 17, 49, 13, 41, 25, 39, 14, 42, 12, 39, 38, 41, 10, 46, 11, 41,
> 22, 41, 10, 42, 12, 38, 26, 36, 14, 46, 11, 44, 21, 40, 16, 41, 14, 41, 27,
> 37, 11, 49, 13, 36, 31, 41, 13, 46, 10, 46, 26, 41, 8, 43, 16, 36, 28, 36,
> 14, 39, 12, 48, 22, 38, 16, 46, 12, 37, 20, 40}
>
> Enjoy, Zak
>
>
> --- On Sun, 3/29/09, Jonathan Post <jvospost3 at gmail.com> wrote:
>
>> From: Jonathan Post <jvospost3 at gmail.com>
>> Subject: [seqfan] Re: conjectured largest number which is not the sum of an n-almost prime and a prime.
>> To: "Sequence Fanatics Discussion list" <seqfan at list.seqfan.eu>
>> Date: Sunday, March 29, 2009, 8:34 PM
>> I accept all of Franklin T.
>> Adams-Watters corrections and warnings. I
>> did not submit the seq yet because it does need to be
>> checked more
>> deeply by someone with a better platform than I have.
>>
>> There's certainly a lot of dithering even at the start of
>> underlying
>> seqs, such as:
>>
>> Number of distinct ways to express n as the sum of a
>> semiprime and a prime:
>>
>> n  a(n)
>> 1   0
>> 2   0
>> 3   0
>> 4   0
>> 5   0
>> 6   1
>> 7   1
>> 8   1
>> 9   2
>> 10   0
>> 11  3
>> 12  2
>> 13  2
>> 14  1
>> 15  3
>> 16  2
>> 17  5
>> 18  1
>> 19  2
>> 20  2
>> 21  3
>> 22  2
>> 23  4
>> 24  2
>> 25  3
>> 26  3
>> 27  5
>> 28  5
>> 29  4
>> 30  1
>> 31  2
>> 32  4
>> 33  5
>> 34  2
>> 35  4
>> 36  3
>> 37  5
>> ...
>>
>> On Sun, Mar 29, 2009 at 5:23 PM,  <franktaw at netscape.net>
>> wrote:
>> > If this is submitted, the title should be "Largest
>> number which is not
>> > the sum of an n-almost prime and a prime", not
>> "Conjectured largest
>> > ....".  That all the values are conjectured belongs
>> in a comment.
>> >
>> > I would be reluctant to conjecture the value of a(6)
>> below without
>> > checking past 10^9 -- probably at least to 10^10.
>>  Large gaps near the
>> > end of these sequences seem to be the rule rather than
>> the exception.
>> >
>> > Franklin T. Adams-Watters
>> >
>> >
>> > -----Original Message-----
>> > From: Jonathan Post <jvospost3 at gmail.com>
>> >
>> > Connecting these seqs:
>> > A130588      Integers which are not the sum of a
>> 3-almost prime and a
>> > prime.
>> > A146295      Integers which are not the sum of a
>> 4-almost prime and a
>> > prime.
>> > A146296      Integers which are not the sum of a
>> 5-almost prime and a
>> > prime.
>> > A146297      Integers which are not the sum of a
>> 6-almost prime and a
>> > prime.
>> >
>> > we get the following... ALL of whose values are
>> conjectures, although
>> > checked through 10^9
>> >
>> > a(n) is the conjectured largest number which is not
>> the sum of an
>> > n-almost prime and a prime.
>> > n=2,3,4,5,6: a(n) = 10, 300, 60060, 3573570,
>> 446185740
>> >
>> > These are not as artificial as they seem, as they
>> derive from Chen's
>> > approach to proving Goldbach's conjecture.