about this summer

[N. J. A. Sloane]

spare time.  The main thing is to add important new sequences,
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Subject: Re: First 11 values of A121387 coincide with apparently unrelated new seq
Date: Fri, 22 Jun 2007 15:24:39 -0400
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So the correspondence is NOT a coincidence.

It is well known that Pythagorean triples can be parameterized as:

x^2 - y^2
2xy
x^2 + y^2;

the triple is primitive if x and y are relatively prime and not both 
odd.

The x^2 + y^2 is the hypotenuse.  It is also well known that a number 
is the sum of two squares iff every prime divisor = 3 (mod 4) is 
present an even number of times.  So if the x^2 + y^2 is a semiprime, 
it is either the product of two primes = 1 (mod 4), or the square of a 
prime = 3 (mod 4).  It is not hard to show that the latter case is 
exclusively x = p and y = 0, which are not relatively prime.  So the 
semiprime Pythagorean triple hypotenuses are exactly the products of 
two primes = 1 (mod 4) - hence A121387, once that is corrected.

Franklin T. Adams-Watters

-----Original Message-----
From: g_m at mcraefamily.com <g_m at mcraefamily.com>

Sequence A121387 needs to be corrected to remove 329, which, unlike all 
the
other elements of this sequence, is NOT the product of two primes of the
form 4n+1.

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