Let us denote with
It is known that every odd prime divides the repunit .
R.Francis & T.Ray call a composite number deceptive
if it has the same property, i.e., if it divides the repunit .
For example, is deceptive because it divides .
Francis & Ray have proved that there are infinite deceptive numbers
since, if is deceptive, then is deceptive as well.
Every number greater than 2980 can be written as the sum of deceptive numbers.
The first deceptive numbers are
91, 259, 451, 481, 703, 1729, 2821, 2981, 3367, 4141, 4187, 5461, 6533, 6541, 6601, 7471, 7777, 8149, 8401 more terms
Pictorial representation of remainders (mod 2, 3, ...,11) frequency. For a table of values and more details click here
A graph displaying how many deceptive numbers are multiples of the primes p
from 2 to 71. In black the ideal line 1/p