##
Explanations.

For a given strictly increasing infinite sequence f = (f_{1},f_{2},...) of positive integers

we determine the largest integer which is not representable as a sum of distinct members of the sequence f.

If it exists, this integer is called the **threshold of completeness** t(f) of the sequence f. For example

t(f) = **-1** for the sequence f of all powers of 2,

t(f) = **-1** for the sequence f of all positive integers,

t(f) = **128** for the sequence f of all squares,

t(f) = **12758** for the sequence f of all cubes [1,2,3],

t(f) = **5134240** for the sequence f of all 4^{th} powers [3], without knowledge of [1],

t(f) = **67898771** for the sequence f of all 5^{th} powers

(discovered **March, 01, 2009**, proved **May, 23, 2009**, without knowledge of [4]),

t(f) = **infinity** for the sequence f of all powers of 3.

The present **Java applet** is designed for sequences of powers,

either with fixed exponent n >= 1 and variable basis x >= 1, i.e., f = (x^{n},(x+1)^{n},...) (check box **Exponential** disabled).

or with fixed basis a >= 2 and variable exponent x >= 0, i.e., f = (a^{x},a^{x+1},...) (check box **Exponential** activated).

The check box **Function** should only be activated for a modest **Number of Recursion Steps**.

**Insider Know How:**

Instead of storing the **characteristic function** of representable integers in a file of bytes [3] on the hard disk

we make use of the Java class java.math.**BigInteger** as an **infinitely extensible bit list** in the main memory.

In each recursion step,
the method **shiftLeft()** is used for a **translation**
and the method **or()** for a **superposition** of characteristic functions.

**Warning.**
For the sequence of 5^{th} powers, we expect a storage capacity of more than 10MB for the bit lists.

Latest Result (**May, 23, 2009**):
For the sequence of 5^{th} powers, the Java Virtual Machine (VM) has to be tuned at least by the option **-Xmx512m**.

##
Bibliography.

[1] S. Lin,

**Computer experiments on sequences which form integral bases**,

pp. 365-370 of J. Leech, editor, Computational Problems in Abstract Algebra. Pergamon, Oxford, 1970.

[2] R. E. Dressler and T. Parker,

**"12,758"**,

Math. Comp., **28** (1974), 313 - 314.

[3] Daniel C. Mayer,

**Sharp bounds for the partition function of integer sequences**,

BIT **27** (1987), 98 - 110.

[4] Harry L. Nelson,

**The Partition Problem**,

J. Rec. Math., **20** (1988), 315 - 316.

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Algebra.