Parasitic number

An n-parasitic number for is a natural number such that, when multiplied by n, the decimal representation of the result is the same as for the original number, except with the rightmost digit moved to the front. In other words, the decimal representation undergoes a right circular shift by one place.

An n-parasitic number can be found as follows. If m is the order of 10 modulo (10n − 1), then

is an n-parasitic number. For instance, if n = 2, then 10n − 1 = 19, and the repeating decimal for 1/19 is

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

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

The smallest n-parasitic numbers are:

n	Smallest n-parasitic number
2	105263157894736842
3	1034482758620689655172413793
4	102564
5	142857
6	10169491525423728813559932203389830508474576271186440677966
7	1014492753623188405797
8	1012658227848
9	10112359550561797752808988764044943820224719

The formula given above does not generate all the elements of the above table; it instead generates 1020408163265030612244897959183673469387755 as the 5-parasitic number. The other n-parasitic numbers generated by the formula are the smallest for their n.

References

* C. A. Pickover, Wonders of Numbers, Chapter 28, Oxford University Press UK, 2000.

* Sequence A092697 in the On-Line Encyclopedia of Integer Sequences.

Number Theory