m=p^2+q^2, (p<q primes): number of presentations

[zak seidov]

to: MH, RM, Neil, seqfans
subj: m=p^2+q^2, (p<q primes) & A088919 
 
O my God,
this is what i shouldn't have as "my last one" -
as great Garry Kasparov did 
loosing hopelessly his last game to Topalov.

I was so firmly sure that my "presentations" are new
that even didn't care to check OEIS database.

Anyway another good lesson,
and thank you all responding
for your patience and kindness
(not mentioning your knowledge
which is well above my head).
God bless you all,
zak

--- Maximilian Hasler <maximilian.hasler at gmail.com>
wrote:

> this is:
> 
> A088919 		Smallest number having exactly n
> representations as sum of
> two squares of distinct primes.
> 	1, 13, 410, 2210, 10370, 202130, 229970, 197210,
> 81770, 18423410,
> 16046810, 12625730, 21899930, 9549410 (list; graph;
> listen)
> 
> Maximilian
> 
> On Tue, Feb 26, 2008 at 3:21 PM, zak seidov
> <zakseidov at yahoo.com> wrote:
> > Dear seqfans,
> >
> >  first of all,
> >  many thanks to you those responding
> >  (and sorry to much more those annoying..)
> >  to my question_posts.
> >
> >  Here is my last one.
> >  Can anyone plz check and extend this?
> >  thanks, zak
> >
> >
> >  A1:
> >  Smallest positive number expressible as sum of
> squares
> >  of two distinct primes p<q exactly in n ways:
> >  1, 13, 410, 2210, 10370, 202130, 229970, 197210,
> 81770
> >  0 way: 1
> >  1 way: 13=2^2+3^2,
> >  2 ways: 410=7^2+19^2=11^2+17^2,
> >  3 ways: 2210=19^2+43^2=23^2+41^2=29^2+37^2,
> >  4 ways:
> >  10370=13^2+101^2=31^2+97^2=59^2+83^2=71^2+73^2,
> >  5 ways:
> > 
>
202130=23^2+449^2=97^2+439^2=163^2+419^2=211^2+397^2=251^2+373^2,
> >
> >  6 ways:
> > 
>
229970=23^2+479^2=109^2+467^2=193^2+439^2=263^2+401^2=269^2+397^2=331^2+347^2,
> >  7 ways:
> > 
>
197210=31^2+443^2=67^2+439^2=107^2+431^2=173^2+409^2=199^2+397^2=241^2+373^2=311^2+317^2,
> >  8 ways:
> > 
>
81770=41^2+283^2=53^2+281^2=71^2+277^2=97^2+269^2=137^2+251^2=157^2+239^2=179^2+223^2=193^2+211^2,
> >  9 ways: ?
> >  10 ways: ?
> >  ...........
> >  ...........
> >
> >  BTW for those interested:
> >  for m's up to 10^6,
> >  there are only 8732 m's
> >  expressible as p^2+q^2 (p<q primes)
> >  at least in 1 way,
> >  right?
 



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I find disparity between my version of the Engel expansion
of the Thue-Morse constant A014571
0.4124540336401075977833613682584552830894783744557695575733794153487935923....
and the terms in A096394, problems starting with A096394(17). This seems to
be associated with a difference between my Thue-Morse constant and A014571(39)
as marked above.

My version is
A096394 := [3, 5, 6, 9, 12, 19, 92, 173, 242, 703, 1861, 3186, 4746, 7843,






Any independent calculation ?
Richard Mathar, http://www.strw.leidenuniv.nl/~mathar