[seqfan] Re: Is A026477 determined by prime signatures?
Bob Selcoe
rselcoe at entouchonline.net
Fri Aug 26 23:21:30 CEST 2016
Hi Graeme and Seqfans,
I submitted a proof yesterday (revised today) on A026477 for Charles'
initial conjecture, noting that defining which signatures are IN and OUT
becomes increasingly complex as the sequence progresses; I welcome any
revisions to make the proof clearer.
>Then, find the "smallest" prime signature not previously listed as IN or
> OUT. This, of course, is the next one IN the sequence. The "smallest"
> prime
> signature means the prime signature with the smallest sum, listed in
> high-to-low collating sequence. (e.g. prime signatures whose sum is 4
> would
> be considered in this order: (4), (3,1), (2,2), (2,1,1), (1,1,1,1))
Graeme, if I understand correctly, I think you may encounter some difficulty
with this approach. For example, (5,5) is OUT because (3,3) and (2) are IN
rather than (2,2). So I think it's more complicated than just looking at
signature sums. Am I missing something?
Overall, I suspect coming up with a solution will be quite challenging.
Cheers,
Bob S.
--------------------------------------------------
From: "Graeme McRae" <graememcrae at gmail.com>
Sent: Friday, August 26, 2016 2:26 PM
To: "Sequence Fanatics Discussion list" <seqfan at list.seqfan.eu>
Subject: [seqfan] Re: Is A026477 determined by prime signatures?
> I've been reading this thread about A026477 with great interest, and have
> been trying to think about how I would write a program to build a list of
> prime signatures representing numbers that are IN (and a list that are
> OUT)
> of the sequence.
>
> My first thought was to start with the prime signatures of 1, 2, and 3,
> which are (), (1), and (1), respectively, and then consider the "products"
> of all triples of prime signatures that are already IN. These products,
> then, will be OUT.
>
> Then, find the "smallest" prime signature not previously listed as IN or
> OUT. This, of course, is the next one IN the sequence. The "smallest"
> prime
> signature means the prime signature with the smallest sum, listed in
> high-to-low collating sequence. (e.g. prime signatures whose sum is 4
> would
> be considered in this order: (4), (3,1), (2,2), (2,1,1), (1,1,1,1))
>
> When finding the products of prime signatures, I need to consider that (1)
> and (1) must represent different primes, so their product can't be (2).
> However, the prime factors of (1,1) and (1,1) need not be unique, so their
> products include (2,1,1) and (1,1,1,1) but not (2,2).
>
> I'll give this some more thought, and try a little programming using the
> VBA that comes with MS Excel. Who knows? Maybe some interesting sequences
> (or a comment on this sequence) might come out of it.
>
> --Graeme McRae
> Palmdale, CA
>
> On Fri, Aug 26, 2016 at 11:55 AM, Charles Greathouse <
> charles.greathouse at case.edu> wrote:
>
>> I also haven't found a good way of discovering which prime signatures are
>> in the sequence. In principle this is combanatorial but I don't know of a
>> good algorithm.
>>
>> Charles Greathouse
>> Case Western Reserve University
>>
>> On Fri, Aug 26, 2016 at 7:17 AM, Don Reble <djr at nk.ca> wrote:
>>
>> > A026477... a(1) = 1, a(2) = 2, a(3) = 3; and for n > 3,
>> >> a(n) = smallest number > a(n-1) and not of the form a(i)*a(j)*a(k)
>> >> for 1 <= i < j < k < n.
>> >>
>> >> It seems that if two numbers have the same prime signature (multiset
>> >> of
>> >> prime exponents) then either both or neither are in the sequence, but
>> >> I
>> >> can't prove this.
>> >>
>> >
>> > Just do strong induction on the number of prime factors (sum of
>> > signature exponents).
>> >
>> > ... prime powers p^r can only be r = {1,2,4,8,15,22...}, ...
>> >>
>> >
>> > Yes: A026474.
>> > Also, square-free numbers have 3n+1 prime factors.
>> >
>> > This suggests that A026477 intersect A025487 (least value of each
>> > signature) would be a worthy sequence. But I don't see how to easily
>> > recognize those signatures.
>> >
>> > value signature
>> > 1:
>> > 2: 1
>> > 4: 2
>> > 16: 4
>> > 120: 3 1 1
>> > 210: 1 1 1 1
>> > 216: 3 3
>> > 256: 8
>> > 384: 7 1
>> > 2880: 6 2 1
>> > 6300: 2 2 2 1
>> > 7200: 5 2 2
>> > 15360: 10 1 1
>> > 15552: 6 5
>> > 26880: 8 1 1 1
>> > 27648: 10 3
>> > 32768: 15
>> > 49152: 14 1
>> > 73728: 13 2
>> > 83160: 3 3 1 1 1
>> > 120120: 3 1 1 1 1 1
>> > 189000: 3 3 3 1
>> > 510510: 1 1 1 1 1 1 1
>> > 921600: 12 2 2
>> > 1399680: 7 7 1
>> > 1966080: 17 1 1
>> > 2365440: 11 1 1 1 1
>> > 2822400: 8 2 2 2
>> > 2985984: 12 6
>> > 3440640: 15 1 1 1
>> > 4194304: 22
>> > 4860000: 5 5 4
>> > 5670000: 4 4 4 1
>> > 6291456: 21 1
>> > 6912000: 11 3 3
>> > 9437184: 20 2
>> > 10644480: 10 3 1 1 1
>> > 15375360: 10 1 1 1 1 1
>> > 60466176: 10 10
>> > 65345280: 8 1 1 1 1 1 1
>> > 71663616: 15 7
>> > 117964800: 19 2 2
>> > 127401984: 19 5
>> > 161243136: 13 9
>> > 251658240: 24 1 1
>> > 251942400: 9 9 2
>> > 302776320: 18 1 1 1 1
>> > 361267200: 15 2 2 2
>> > 440401920: 22 1 1 1
>> > 536870912: 29
>> > 805306368: 28 1
>> > 892371480: 3 1 1 1 1 1 1 1 1
>> > 1109908800: 6 2 2 2 2 1
>> > 1207959552: 27 2
>> > 1327104000: 17 4 3
>> > 1968046080: 17 1 1 1 1 1
>> > 4232632320: 11 10 1 1
>> > 6469693230: 1 1 1 1 1 1 1 1 1 1
>> > 9172942848: 22 7
>> > 9932482560: 11 1 1 1 1 1 1 1
>> > 10883911680: 12 12 1
>> > 12570798240: 5 5 1 1 1 1 1 1
>> > 13759414272: 21 8
>> > 13783770000: 4 4 4 1 1 1 1
>> > 15330615300: 2 2 2 2 2 2 1
>> > 16307453952: 26 5
>> > 23279477760: 10 10 1 1 1
>> > 24461180928: 25 6
>> > 32212254720: 31 1 1
>> > 32248627200: 16 9 2
>> > 38755368960: 25 1 1 1 1
>> > 39729690000: 4 4 4 3 1 1
>> > 56371445760: 29 1 1 1
>> > 68719476736: 36
>> > 103079215104: 35 1
>> > 114223549440: 10 1 1 1 1 1 1 1 1
>> > 154618822656: 34 2
>> > 156728328192: 15 14
>> > 169869312000: 24 4 3
>> > 251909898240: 24 1 1 1 1 1
>> > 408410100000: 5 5 5 5
>> > 717001084800: 7 2 2 2 2 1 1 1
>> > 812665405440: 17 11 1 1
>> > 828120733440: 8 1 1 1 1 1 1 1 1 1
>> >
>> > --
>> > Don Reble djr at nk.ca
>> >
>> >
>> >
>> > --
>> > Seqfan Mailing list - http://list.seqfan.eu/
>> >
>>
>> --
>> Seqfan Mailing list - http://list.seqfan.eu/
>>
>
> --
> Seqfan Mailing list - http://list.seqfan.eu/
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