A097101-3: insert n^2 and divide by 5(?)

[zak seidov]

Neil, Jim,seqfans,

i guess that "n^2" should be inserted after "i.e." in
these A's.

Also, are all terms divisible by 5?

zak

A097101  Numbers n that are the hypotenuse of exactly
7 distinct integer sided right triangles, i.e. can be
written as a sum of two squares in 7 ways.   
  325, 425, 650, 725, 845, 850, 925, 975, 1025, 1275, 

A097102  Numbers n that are the hypotenuse of exactly
13 distinct integer sided right triangles, i.e. can be
written as a sum of two squares in 13 ways.    
 1105, 1885, 2210, 2405, 2465, 2665, 3145, 3315, 

A097103  Numbers n that are the hypotenuse of exactly
22 distinct integer sided right triangles, i.e. can be
written as a sum of two squares in 22 ways.   
 5525, 9425, 11050, 12025, 12325, 13325, 14365, 
 
 
 

 


Jim Buddenhagen (jbuddenh(AT)gmail.com), 


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zs> From seqfan-owner at ext.jussieu.fr  Tue Feb 26 07:51:23 2008
zs> Date: Mon, 25 Feb 2008 22:51:01 -0800 (PST)
zs> From: zak seidov <zakseidov at yahoo.com>
zs> Subject: A097101-3: insert n^2 and divide by 5(?)
zs> To: njas at research.att.com, jbuddenh at gmail.com, seqfan at ext.jussieu.fr
zs> 
zs> Neil, Jim,seqfans,
zs> 
zs> i guess that "n^2" should be inserted after "i.e." in
zs> these A's.
zs> 
zs> Also, are all terms divisible by 5?
zs> 
zs> zak
zs> 
zs> A097101  Numbers n that are the hypotenuse of exactly
zs> 7 distinct integer sided right triangles, i.e. can be
zs> written as a sum of two squares in 7 ways.
zs>   325, 425, 650, 725, 845, 850, 925, 975, 1025, 1275,
zs> 
zs> A097102  Numbers n that are the hypotenuse of exactly
zs> 13 distinct integer sided right triangles, i.e. can be
zs> written as a sum of two squares in 13 ways.
zs>  1105, 1885, 2210, 2405, 2465, 2665, 3145, 3315,
zs> 
zs> A097103  Numbers n that are the hypotenuse of exactly
zs> 22 distinct integer sided right triangles, i.e. can be
zs> written as a sum of two squares in 22 ways.
zs>  5525, 9425, 11050, 12025, 12325, 13325, 14365,

Zak's interpretation is confirmed with A097101:
the smallest number that can be written as a sum of two nonzero
1296+ 104329= 105625=325^2
6400+ 99225= 105625=325^2
8281+ 97344= 105625=325^2
15625+ 90000= 105625=325^2
27225+ 78400= 105625=325^2
38025+ 67600= 105625=325^2
41616+ 64009= 105625=325^2

The second value is 180625=425^2. So the appropriate formula line in A097101 is
A097101 = {n: A025426(n^2)=7}
and the definition is indeed "Numbers n that are the hypotenuse of exactly 7 distinct
integer sided right triangles, ie, n^2 can be written as a sum of two non-zero

Now this comes with some sort of shortcoming which deserves an additional
comment in these sequences:
there are non-square x which can be written as a sum of 2 non-zero
203125 = 125*sqrt(13) is the first example,

265625 = 125*sqrt(17) the second,
406250=125*sqrt(26) the third,
which I detect (plz check). The next value is 422500=650^2, which is a perfect
The naive expectation that all solutions to A025426(x)=7 are found by collecting
[A097101(n)]^2 seems not to be met.
So if someone has some computer power available, it probably would make sense
to gather the first few of the (square or nonsquare!) x with
A025426(x)=7, and do the equivalent for the other sequences of that type.
In addition, adding cross-references to A025426 which say where the first
occurrences (records of A025426) are would also make sense.

I have no statistics on the other aspect (are these solutions necessarily multiples
of 5 or 25?).

Richard