Parasitic number

An n-parasitic number (in base 10) is a positive natural number which can be multiplied by n by moving the rightmost digit of its decimal representation to the front. Here n is itself a single-digit positive natural number. In other words, the decimal representation undergoes a right circular shift by one place. For example, 4•128205=512820, so 128205 is 4-parasitic. Most authors do not allow leading zeros to be used, and this article follows that convention. So even though 4•025641=102564, the number 025641 is not 4-parasitic.

Derivation

An n-parasitic number can be derived by starting with a digit k (which should be equal to n or greater) in the rightmost (units) place, and working up one digit at a time. For example, for n = 4 and k = 7:

4•7=28

4•87=348

4•487=1948

4•9487=37948

4•79487=317948

4•179487=717948.

So 179487 is a 4-parasitic number with units digit 7. Others are 179487179487, 179487179487179487 etc.

Notice that the repeating decimal

$x=0.179487179487179487\ldots=0.\overline{179487} \mbox{ has }4x=0.\overline{717948}=\frac{7.\overline{179487}}{10}.$

Thus

$4x=\frac{7+x}{10} \mbox{ so } x=\frac{7}{39}.$

In general, an n-parasitic number can be found as follows. Pick a one digit integer k such that k ≥ n, and take the period of the repeating decimal k/(10n−1). This will be $\frac{k}{10n-1}(10^m-1)$ where m is the length of the period; i.e. the order of 10 modulo (10n − 1).

For another example, if n = 2, then 10n − 1 = 19 and the repeating decimal for 1/19 is

$\frac{1}{19}=0.\overline{052631578947368421}.$

So that for 2/19 is double that:

$\frac{2}{19}=0.\overline{105263157894736842}.$

The length m of this period is 18, the same as the order of 10 modulo 19, so 2 × (10¹⁸ − 1)/19 = 105263157894736842.

105263157894736842 × 2 = 210526315789473684, which is the result of moving the last digit of 105263157894736842 to the front.

Smallest n-parasitic numbers

The smallest n-parasitic numbers are also known as Dyson numbers, after a puzzle concerning these numbers posed by Freeman Dyson.^[1][2][3] They are:

n	Smallest n-parasitic number	period of
1	1	1/9
2	105263157894736842	2/19
3	1034482758620689655172413793	3/29
4	102564	4/39
5	142857	7/49=1/7
6	10169491525423728813559932203389830508474576271186440677966	6/59
7	1014492753623188405797	7/69
8	1012658227848	8/79
9	10112359550561797752808988764044943820224719	9/89

General note

In general, if we relax the rules to allow a leading zero, then there are 9 n-parasitic numbers for each n. Otherwise only if k ≥ n then the numbers do not start with zero and hence fit the actual definition.

Other n-parasitic integers can be built by concatenation. For example, since 179487 is a 4-parasitic number, so are 179487179487, 179487179487179487 etc.

See also

• Cyclic number

Notes

1. ^ Dawidoff, Nicholas (March 25, 2009), "The Civil Heretic", New York Times Magazine, http://www.nytimes.com/2009/03/29/magazine/29Dyson-t.html .
2. ^ Tierney, John (April 6, 2009), "Freeman Dyson’s 4th-Grade Math Puzzle", New York Times, http://tierneylab.blogs.nytimes.com/2009/04/06/freeman-dysons-4th-grade-math-puzzle/ .
3. ^ Tierney, John (April 13, 2009), "Prize for Dyson Puzzle", New York Times, http://tierneylab.blogs.nytimes.com/2009/04/13/prize-for-dyson-puzzle/ .

References

• C. A. Pickover, Wonders of Numbers, Chapter 28, Oxford University Press UK, 2000.
• Sequence  A092697 in the On-Line Encyclopedia of Integer Sequences.