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.
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
So 179487 is a 4-parasitic number with units digit 7. Others are 179487179487, 179487179487179487 etc.
Notice that the repeating decimal
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 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
So that for 2/19 is double that:
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
|n||Smallest n-parasitic number||period of|
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.
- 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