5 + 3 = 8
Marc LeBrun
mlb at well.com
Fri Sep 21 17:33:40 CEST 2001
"Five plus three equals eight" opens Philibert Schogt's "The Wild Numbers"
(http://www.amazon.com/exec/obidos/ASIN/0452282470) and so we know that
the Diophantine equations
X + 3 = 8
X + 5 = 8
have solutions (namely X = 5 and X = 3 respectively).
Moreover
X + 2 = 8
X + 6 = 8
2 X^2 = 8
X^2 + X + 2 = 8
(for X = 6 once and X = 2 thrice)... and how many others?
(To stay finite, we'll restrict these to "power partition polynomials" with
non-negative coefficients, and roots X greater than one.)
My (hopefully debugged) program counted seventeen distinct expressions like
this for n = 8, while generating the new sequence A064572:
0 1 2 5 6 10 11 17 20 26 27 38 39 47 51 65 66 82 83 102 107 123 124 153...
Thinking there ought to be some multiplicative pattern, I then restricted
the count to just polynomials in prime X, and got A064573:
0 1 2 4 5 8 9 13 15 20 21 29 30 37 40 50 51 64 65 80 84 99 100 123 125...
which of course has the composite X complement A064574 (= A064572 -
A064573):
0 0 0 1 1 2 2 4 5 6 6 9 9 10 11 15 15 18 18 22 23 24 24 30 31 32 34 38...
Note that the number of solutions for prime n seems to increase by only one.
This sort of makes sense; it's the minimum increase: you just add 1 to all
the previous expressions for n-1, then adjoin the single expression X = n.
(But why?)
Anyhow this suggested taking the first differences of the above three
sequences, giving A064575, A064576, and A064577:
1 1 3 1 4 1 6 3 6 1 11 1 8 4 14 1 16 1 19 5 16 1 29 3 22 7 31 1 37 1...
1 1 2 1 3 1 4 2 5 1 8 1 7 3 10 1 13 1 15 4 15 1 23 2 21 5 27 1 33 1 36...
0 0 1 0 1 0 2 1 1 0 3 0 1 1 4 0 3 0 4 1 1 0 6 1 1 2 4 0 4 0 6 1 1 1 9...
But neither I nor Superseeker could get anywhere formulating any of these
(including several variations).
The only "hit" was that the last sequence above happens to initially match
A028422[n+1]:
0 0 1 0 1 0 2 1 1 0 3 0 1 1 4 0 3 0 3 1 1 0 6 1 1 2 3 0 4 0 6 1 1 1 8...
"Number of ways n can be properly factored." up to n=19, after which they
diverge (though rather delicately; there's a lot of matching terms that
follow). Are these related? Can their difference be characterized?
I'm surprised that these sequences seem to be neither known nor obviously
related to anything in the EIS. It's possible that my definition could be
diddled somehow to get something recognizable. For example in submitting
these I just tersely described them as "partitions into powers of X" but,
strictly speaking, I suppose that's quibblable (eg with regard to
representations of unity).
So far, it looks like the most I might presume is that, say, a(239) =
a(238)+1. But I'd still have to compute a(238), and neither value
apparently helps me with a(240).
Is there a way to compute these values more directly than by enumerating the
partitions?
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