A036236 and A078457

[Hans Havermann]

still there: Entropy!
says nothing about existence, since a(n) = infinity is implied  
shorthand for non-existence.
street) here:
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Date: Thu, 17 Jul 2008 11:56:19 -0700
From: "Jonathan Post" <jvospost3 at gmail.com>
To: SeqFan <seqfan at ext.jussieu.fr>
Subject: Sequence suggested: sum of at most 4 nonzero 4-th powers in more than one way
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I'm looking to make a sequence derived as a special subset of A004833
Numbers that are the sum of at most 4 nonzero 4-th powers.

This would be "numbers that are the sum of (exactly) or (no more than)
4 nonzero 4-th powers in more than one way."

This can happen in 4 ways, as exemplified in
Piezas, Tito III and Weisstein, Eric W. "Diophantine Equation--4th
Powers." From MathWorld--A Wolfram Web Resource.
http://mathworld.wolfram.com/DiophantineEquation4thPowers.html

(i) A^4 + B^4 + C^4 + D^4 = E^4  (the 4.1.4 equation).
Smallest example is 30^4 + 120^4 + 272^4 + 315^4 = 353^4 = 15527402881.

(ii) A^4 + B^4 + C^4 + D^4 = E^4 + F^4  (the 4.2.4 equation).
Parametric solutions are known. Smallest example is
5^4 + 5^4 + 6^4 + 8^4 = 3^4 + 9^4 = 6642  (Ramanujan)
If we drop the "nonzero" we have
3^4 + 5^4 + 8^4 + 0^4 = 7^4 + 7^4 = 4802  (Ramanujan)

(iii) A^4 + B^4 + C^4 + D^4 = E^4 + F^4 + G^4  (the 4.3.4 equation).
Parametric solutions are known. Smallest example is
4^4 + 4^4 + 5^4 + 6^4 = 2^4 + 2^4 + 7^4 = 2433 (Ramanujan).
Ramanujan also gave:
7^4 + 8^4 + 10^4 + 13^4 = 3^4 + 9^4 + 14^4 = 45058.
5^4 + 5^4 + 6^4 + 14^4 = 7^4 + 10^4 + 13^4 = 40962.

(iv) A^4 + B^4 + C^4 + D^4 = E^4 + F^4 + G^4 + H^4  (the 4.4.4 equation).
I don't know the smallest.

If we drop the "exactly" for the "at most" we have values
corresponding to solutions
A^4 + B^4 + C^4 = D^4 (4.1.3 equation)
Cf. (N. Elkies, 1987) and (Roger Frye, 1988) originally thought
nonexistent (also known as the Euler quartic conjecture).

and solutions of A^4 + B^4 = C^4 + D^4 (4.2.2 equation)
Cf. A003824, A018786.
A003824  Sum of two 4th powers in more than one way (primitive solutions).  	 	
635318657, 3262811042, 8657437697, 68899596497, ...
A018786  Sum of two 4th powers in more than one way.
635318657, 3262811042, 8657437697, 10165098512, ...

So is there an interesting sequence which begins:
2433, 4802, 6642?

If so, what is a(4)?




If the sequence A139334 is continued following the rules in the definition, we get:
2, 24, 28, 311, 312, 325, 337, 340, 365, 398, 405, 439,...
with the first differences A139334:
The problem that arises is: the next difference, a139334(12) requires a leading 0
to match the final 0 in a(8)=340; this cannot be included in the OEIS.

I have no access to the Angelini article and cannot say how it works around that
case.

Richard