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 multiplicative 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: (leading zeros are not allowed) (sequence A092697 in OEIS)

n	Smallest n-parasitic number	Digits	Period of
1	1	1	1/9
2	105263157894736842	18	2/19
3	1034482758620689655172413793	28	3/29
4	102564	6	4/39
5	142857	6	7/49 = 1/7
6	1016949152542372881355932203389830508474576271186440677966	58	6/59
7	1014492753623188405797	22	7/69
8	1012658227848	13	8/79
9	10112359550561797752808988764044943820224719	44	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.

Other bases

In duodecimal system, the smallest n-parasitic numbers are: (using inverted two and three for ten and eleven, respectively) (leading zeros are not allowed)

n	Smallest n-parasitic number	Digits	Period of
1	1	1	1/Ɛ
2	10631694842	Ɛ	2/1Ɛ
3	2497	4	7/2Ɛ = 1/5
4	10309236ᘔ88206164719544	1Ɛ	4/3Ɛ
5	1025355ᘔ9433073ᘔ458409919Ɛ715	25	5/4Ɛ
6	1020408142854ᘔ997732650ᘔ18346916306	2Ɛ	6/5Ɛ
7	101899Ɛ864406Ɛ33ᘔᘔ15423913745949305255Ɛ17	35	7/6Ɛ
8	131ᘔ8ᘔ	6	ᘔ/7Ɛ = 2/17
9	101419648634459Ɛ9384Ɛ26Ɛ533040547216ᘔ1155Ɛ3Ɛ12978ᘔ399	45	9/8Ɛ
ᘔ	14Ɛ36429ᘔ7085792	14	12/9Ɛ = 2/15
Ɛ	1011235930336ᘔ53909ᘔ873Ɛ325819Ɛ9975055Ɛ54ᘔ3145ᘔ42694157078404491Ɛ	55	Ɛ/ᘔƐ

Strict definition

In strict definition, least number m beginning with 1 such that the quotient m/n is obtained merely by shifting the leftmost digit 1 of m to the right end are

1, 105263157894736842, 1034482758620689655172413793, 102564, 102040816326530612244897959183673469387755, 1016949152542372881355932203389830508474576271186440677966, 1014492753623188405797, 1012658227848, 10112359550561797752808988764044943820224719, 10, 100917431192660550458715596330275229357798165137614678899082568807339449541284403669724770642201834862385321, 100840336134453781512605042016806722689075630252, ... (sequence A128857 in OEIS)

They are the period of n/(10n - 1), also the period of the decadic integer -n/(10n - 1).

Number of digits of them are

1, 18, 28, 6, 42, 58, 22, 13, 44, 2, 108, 48, 21, 46, 148, 13, 78, 178, 6, 99, 18, 8, 228, 7, 41, 6, 268, 15, 272, 66, 34, 28, 138, 112, 116, 179, 5, 378, 388, 18, 204, 418, 6, 219, 32, 48, 66, 239, 81, 498, ... (sequence A128858 in OEIS)

See also

• Cyclic number

Notes

1. Dawidoff, Nicholas (March 25, 2009), "The Civil Heretic", New York Times Magazine.
2. Tierney, John (April 6, 2009), "Freeman Dyson’s 4th-Grade Math Puzzle", New York Times.
3. Tierney, John (April 13, 2009), "Prize for Dyson Puzzle", New York Times.

References

• C. A. Pickover, Wonders of Numbers, Chapter 28, Oxford University Press UK, 2000.
• Sequence  A092697 in the On-Line Encyclopedia of Integer Sequences.
• Bernstein, Leon (1968), "Multiplicative twins and primitive roots", Mathematische Zeitschrift 105: 49–58, doi:10.1007/BF01135448, MR 0225709