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.