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

[Richard Mathar]

squares in 7 different ways is 105625=325^2:
squares in exactly 7 ways"
squares in exactly 7 different ways (so these do not find an echo in A097101):
	625+ 202500= 203125
       10404+ 192721= 203125
       18225+ 184900= 203125
       22500+ 180625= 203125
       62500+ 140625= 203125
       69169+ 133956= 203125
       84100+ 119025= 203125
square and again in A097101.
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Message-ID: <d3dac270802260426q3ec3c90fp958cb34a932cd541 at mail.gmail.com>
Date: Tue, 26 Feb 2008 04:26:26 -0800
From: "Max Alekseyev" <maxale at gmail.com>
To: "zak seidov" <zakseidov at yahoo.com>
Subject: Re: A097101-3: insert n^2 and divide by 5(?)
Cc: njas at research.att.com, jbuddenh at gmail.com, seqfan at ext.jussieu.fr
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Zak,

Your conjecture on divisibility by 5 is false.

Actually, A097101 consists precisely of the numbers of any of the
following forms:
2^(2n) * p^2 * q * t
2^(2n) * p^7 * t
2^(2n+1) * p^6 * t
where p,q are any (distinct) primes = 1 (mod 4), t is any positive
integer whose prime divisors = 3 (mod 4), and n is a non-negative
integers.


A097102 consists precisely of the numbers of any of the following forms:
2^(2n) * p * q * r * t
2^(2n) * p^4 * q * t
2^(2n) * p^13 * t
2^(2n+1) * p^2 * q^2 * t
2^(2n+1) * p^12 * t
where p,q,r are any (distinct) primes = 1 (mod 4), t is any positive
integer whose prime divisors = 3 (mod 4), and n is a non-negative
integers.

Finally, A097103 consists precisely of the numbers of any of the
following forms:
2^(2n) * p^2 * q * r * t
2^(2n) * p^4 * q^2 * t
2^(2n) * p^7 * q * t
2^(2n) * p^22 * t
2^(2n+1) * p^21 * t
where p,q,r are any (distinct) primes = 1 (mod 4), t is any positive
integer whose prime divisors = 3 (mod 4), and n is a non-negative
integers.

All these formulas easily follow from the formula (17) at
http://mathworld.wolfram.com/SumofSquaresFunction.html (actually, the
bottom case of the formula), but please double check as I may have
missed something.

Regards,
Max

On Mon, Feb 25, 2008 at 10:51 PM, zak seidov <zakseidov at yahoo.com> wrote:
> 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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