nada, nada, nada ...
Jon Awbrey
jawbrey at att.net
Sun Jun 10 18:24:09 CEST 2007
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reismann at free.fr wrote:
>
> Hi seqfans,
>
> The numbers are nothing. Nothing by themselves.
> <...>
> PS : If these questions were already studied,
> please indicate to me where I can find informations.
> And of course, all the comments are welcome.
not exactly the same, but some ideas similar to yours
are investigated under the head of generalized primes.
there is at least one old book by rademacher, i think.
ja
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At 11:57 AM 6/10/2007, Jon Schoenfield wrote:
>Thanks! I think you're right. I had looked at a few of the pages
>at the web site Robert Gerbicz mentioned, but I hadn't yet been to
>the FAQ page there, i.e.,
>
>http://euler.free.fr/faq.htm
>
>... and if I had, I think the "1141" near the top of that page
>would've jumped out at me. :-)
>
>That page, after giving the equation showing 1141^6 as the sum of
>seven 6th powers,
The book by Wells in the OEIS data gives the 1141 breakdown, so it
must have been known 20 years ago.
From the definition of A048284:
"For k > A048283(n), there are a(n) numbers expressible as a sum of exactly
k-n k-gonal numbers."
and from the phrasing of the EXAMPLE:
"For k >= 6, the 2 numbers 2k-1 and 5k-4 are the sum of at best k k-gonal
numbers. Hence a(0) = 2.
For k >= 9, the 4 numbers k-1, 2k-2, 4k-4, and 5k-5 are the sum of at best
k-1 k-gonal numbers. Hence a(1) = 4.
For k >= 10, the 6 numbers k-2, 2k-3, 3k-4, 4k-5, 5k-6, and 8k-9 are the sum
of at best k-2 k-gonal numbers. Hence a(2) = 6."
and from the lack of any References, Links, Comments, or Programs in
A048284, it is not entirely clear (to me) that a(n) is not merely a lower
bound.
If it is not just a lower bound, can anyone supply a proof or a reference to
one? (It's not in R. K. Guy's book Unsolved Problems in Number Theory, or in
his Monthly article "Every number is expressible as the sum of how many
polygonal numbers?")
Thanks,
Jonathan
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